ON THE ESSENTIAL LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY IN TERMS OF LOGICAL AND “∧”/LOGICAL OR “∨” RELATIONS: REPORT ON THE OCCASION OF THE PUBLICATION OF THE FOUR MAIN PAPERS ON INTER-UNIVERSAL TEICHMÜLLER THEORY Shinichi Mochizuki September 2023 Abstract. The main goal of the present paper is to give a detailed exposition of the essential logical structure of inter-universal Teichmüller theory from the point of view of the Boolean operators such as the logical AND “∧” and logical OR “∨” operators of propositional calculus. This essential logical structure of inter-universal Teichmüller theory may be summarized symbolically as follows: A B = ˙ B 2 ˙ ... ) A (B 1 ˙ B 2 ˙ ... ˙ B 1  ˙ B 2  ˙ ... ) A (B 1 . . . =⇒ where ˙ denotes the Boolean operator exclusive-OR, i.e., “XOR”; · the ∨” · A, B, B 1 , B 2 , B 1  , B 2  , denote various propositions; · the logical AND “∧’s” correspond to the Θ-link of inter-universal Te- ichmüller theory and are closely related to the multiplicative structures of the rings that appear in the domain and codomain of the Θ-link; ˙ · the logical XOR ∨’s” correspond to various indeterminacies that arise mainly from the log-Kummer-correspondence, i.e., from sequences of it- erates of the log-link of inter-universal Teichmüller theory, which may be thought of as a device for constructing additive log-shells. ˙ This sort of concatenation of logical AND “∧’s” and logical XOR ∨’s” is reminiscent of the well-known description of the “carry-addition” operation on Teichmüller ˙ representatives of the truncated Witt ring Z/4Z in terms of Boolean addition ∨” and Boolean multiplication “∧” in the field F 2 and may be regarded as a sort of “Boolean intertwining” that mirrors, in a remarkable fashion, the “arith- metic intertwining” between addition and multiplication in number fields and local fields, which is, in some sense, the main object of study in inter-universal Te- ichmüller theory. One important topic in this exposition is the issue of “redundant copies”, i.e., the issue of how the arbitrary identification of copies of isomorphic mathematical objects that appear in the various constructions of inter-universal Te- ichmüller theory impacts and indeed invalidates the essential logical structure of inter-universal Teichmüller theory. This issue has been a focal point of funda- mental misunderstandings and entirely unnecessary confusion concerning inter-universal Teichmüller theory in certain sectors of the mathematical community. The exposition of the topic of “redundant copies” makes use of many interesting elementary examples from the history of mathematics. Typeset by AMS-TEX 1 2 SHINICHI MOCHIZUKI Contents: Introduction §1. Summary of non-mathematical aspects for non-specialists §1.1. Publication of [IUTchI-IV] §1.2. Redundancy assertions of the “redundant copies school” (RCS) §1.3. Qualitative assessment of assertions of the RCS §1.4. The importance of extensive, long-term interaction §1.5. The historical significance of detailed, explicit, accessible records §1.6. The importance of further dissemination §1.7. The notion of an “expert” §1.8. Fabricated versions spawn fabricated dramas §1.9. Geographical vs. mathematical proximity §1.10. Mathematical intellectual property rights §1.11. Social mirroring of mathematical logical structure §1.12. Computer verification, mathematical dialogue, and developmental reconstruction §2. Elementary mathematical aspects of “redundant copies” §2.1. The history of limits and integration §2.2. Derivatives and integrals §2.3. Line segments vs. loops §2.4. Logical AND “∧” vs. logical OR “∨” §3. The logical structure of inter-universal Teichmüller theory §3.1. One-dimensionality via identification of RCS-redundant copies §3.2. RCS-redundancy of Frobenius-like/étale-like versions of objects §3.3. RCS-redundant copies in the domain/codomain of the log-link §3.4. RCS-redundant copies in the domain/codomain of the Θ-link §3.5. Gluings, indeterminacies, and pilot discrepancy §3.6. Chains of logical AND relations §3.7. Poly-morphisms and logical AND relations §3.8. Inter-universality and logical AND relations §3.9. Passage and descent to underlying structures §3.10. Detailed description of the chain of logical AND relations §3.11. The central importance of the log-Kummer-correspondence List of Examples: 1.5.1. Irrationality, impartiality, and the Voodoo Hypothesis 1.5.2. The internet/mass media as an apple of discord 1.9.1. The insufficiency of geographical proximity 1.9.2. The remarkable potency of mathematical proximity 1.10.1. The Pythagorean Theorem 1.12.1. Explicit parametrization of Pythagorean triples 2.1.1. False contradiction in the theory of integration 2.2.1. Symmetry properties of derivatives 2.3.1. Endpoints of an oriented line segment 2.3.2. Gluing of adjacent oriented line segments 2.4.1. “∧” vs. “∨” for adjacent oriented line segments 2.4.2. Differentials on oriented line segments LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 3 2.4.3. Representation via subgroup indices of “∧” vs. “∨” 2.4.4. Logical “∧/∨” vs. “narrative ∧/∨” 2.4.5. Numerical representation of “∧” vs. “∨” 2.4.6. Carry operations in arithmetic, geometry, and Boolean logic 2.4.7. The projective line as a gluing of ring schemes along a multiplicative group scheme 2.4.8. Gluings of rings along multiplicative monoids 3.1.1. Elementary models of gluings and intertwinings 3.2.1. Global multiplicative subspaces and bounds on heights 3.2.2. Coricity, symmetry, and commutativity properties of the log-theta-lattice 3.3.1. Classical complex Teichmüller theory 3.3.2. The Jacobi identity for the classical theta function 3.3.3. Theta functions and multiplicative structures 3.5.1. Bounded nature of log-shell automorphism indeterminacies 3.5.2. Examples of gluings 3.8.1. Inevitability of inner automorphism indeterminacies 3.8.2. Inter-universality and the structure of ±ell NF-)Hodge theaters 3.8.3. Truncated vs. profinite Kummer theory and compatibility with the p-adic logarithm 3.8.4. Symmetrizing isomorphisms, truncatibility, and the log-Kummer- correspondence 3.9.1. Categories of open subschemes 3.10.1. Symmetries as a fundamental non-formal aspect of gluings 3.10.2. Chains of logical AND relations via commutative diagrams Introduction In the present paper, we give a detailed exposition of the essential logical structure of inter-universal Teichmüller theory in terms of elementary Boolean operators such as logical AND “∧” and logical OR “∨”. One important topic in this exposition is the issue of “redundant copies”, i.e., the issue of how the arbi- trary identification of copies of isomorphic mathematical objects that appear in the various constructions of inter-universal Teichmüller theory impacts and in- deed invalidates the essential logical structure of inter-universal Teichmüller the- ory. This issue has been a focal point of fundamental misunderstandings and entirely unnecessary confusion concerning inter-universal Teichmüller theory in certain sectors of the mathematical community [cf. the discussion of Examples 2.4.5, 2.4.7, 2.4.8]. We begin, in §1, by reporting on various non-mathematical aspects of the situa- tion surrounding inter-universal Teichmüller theory, such as the issue of “redundant copies”. Perhaps the most central portion of this discussion of non-mathematical as- pects of the situation surrounding inter-universal Teichmüller theory concerns the long-term, historical importance of producing detailed, explicit, mathemati- cally substantive, and readily accessible written documentation of the essential logical structure of the issues under debate [cf. §1.5]. Such written documentation of the essential logical structure of the issues under debate is especially important in situations such as the situation that has arisen surrounding inter-universal Te- ichmüller theory, in which the proliferation of logically unrelated fabricated 4 SHINICHI MOCHIZUKI versions of the theory has led to fundamental misunderstandings and entirely un- necessary confusion concerning inter-universal Teichmüller theory in certain sectors of the mathematical community that are deeply detrimental to the operational normalcy of the field of mathematics [cf. §1.3, §1.8, §1.10, §1.11, §1.12]. This dis- cussion in §1 is supplemented by various interesting historical examples related to the irrationality of square roots of prime numbers [cf. Example 1.5.1], the Pythagorean Theorem [cf. Example 1.10.1], and Pythagorean triples [cf. Example 1.12.1]. We then proceed, in §2, to discuss elementary aspects of the mathematics surrounding the essential logical structure of inter-universal Teichmüller theory. Our discussion of these elementary aspects, which concerns mathematics at the ad- vanced undergraduate or beginning graduate level and does not require any advanced knowledge of anabelian geometry or inter-universal Teichmüller theory, focuses on the close relationship between · integration and differentiation on i.e., so to speak, the “differential geometry” of the real line [cf. §2.1, §2.2, as well as Example 2.4.2], · the geometry of adjacent closed intervals of the real line and the loops that arise by identifying various closed subspaces of such closed intervals [cf. §2.3; Example 2.4.1], and · Boolean operators such as logical AND “∧” and logical OR “∨” [cf. §2.4]. One important unifying theme that relates these seemingly disparate topics is the theme of “carry operations”, which appear in the various arithmetic, geomet- ric [i.e., “gluing”], and Boolean-logical situations discussed in §2 [cf. Example 2.4.6]. On the other hand, from the point of view of arithmetic geometry, the discus- sion of the projective line as a gluing of ring schemes along a multiplicative group scheme given in Example 2.4.7 yields a remarkably elementary qualitative model/analogue of the essential logical structure surrounding the gluing given by the Θ-link in inter- universal Teichmüller theory. Moreover, over the complex numbers, this example of the projective line i.e., which may be visualized as a sphere leads to an inter- esting analogy between the well-known [e.g., especially, in a cartographic context!] metric/geodesic geometry of the sphere with the multiradial representa- tion of the Θ-pilot in inter-universal Teichmüller theory [cf. Example 2.4.7, (v)]. This example of the projective line discussed in Example 2.4.7 may be understood as occupying a special role in the exposition of the present paper in light of the fact that it is more directly related to scheme-theoretic arithmetic geometry than the previously mentioned examples and leads naturally to the subsequent ring-/monoid-theoretic Example 2.4.8, which may literally be regarded, i.e., in a much more rigorous, technical sense, as a sort of miniature qualitative model that is to say, so to speak, a sort of “preview” of the gluing constituted by the Θ-link of inter-universal Teichmüller theory. Finally, this example of the projec- tive line is also of interest in light of the remarkable parallels between the issue of LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 5 “redundant copies” in the context of inter-universal Teichmüller theory and the well-known 19-th century “algebraic truths” versus “geometric fantasies” dis- pute between Weierstrass and Riemann concerning approaches to complex function theory [cf. the discussion of §1.5]. The preparatory topics of §2 lead naturally to the detailed exposition of the essential logical structure of inter-universal Teichmüller theory given in §3. From a strictly rigorous point of view, this exposition assumes a substantial level of knowl- edge and understanding of the technicalities of inter-universal Teichmüller theory [which are surveyed, for instance, in [Alien]], although the essential mathematical content of most of the issues discussed may in fact be understood at the level of the elementary considerations discussed in §2. The essential logical structure of inter-universal Teichmüller theory may be represented symbolically as follows: A B =⇒ A (B 1 ˙ B 2 ˙ . . . ) A (B 1 ˙ B 2 ˙ . . . ˙ B 1  ˙ B 2  ˙ . . . ) =⇒ A (B 1 ˙ B 2 ˙ . . . ˙ B 1  ˙ B 2  ˙ . . . ˙ B 1  ˙ B 2  ˙ . . . ) = .. . ˙ ˙ denotes the Boolean cf. the discussion of (∧( ∨)-Chn) in §3.10. [Here, ∨” operator exclusive-OR, i.e., “XOR”.] Indeed, §3 is devoted, for the most part, to giving a detailed exposition of various aspects of this symbolic representation, such as the following: · the logical AND “∧’s” in the above display may be understood as corresponding to the Θ-link of inter-universal Teichmüller theory and are closely related to the multiplicative structures of the rings that appear in the domain and codomain of the Θ-link; ˙ · the logical XOR ∨’s” in the above display may be understood as corresponding to various indeterminacies that arise mainly from the log- Kummer-correspondence, i.e., from sequences of iterates of the log- link of inter-universal Teichmüller theory, which may be thought of as a device for constructing additive log-shells. ˙ This appearance of logical AND “∧’s” and logical XOR ∨’s” is of interest in that it is reminiscent of the well-known description of the “carry-addition” operation on Teichmüller representatives of the truncated Witt ring Z/4Z in terms of Boolean ˙ and Boolean multiplication “∧” in the field F 2 and may be re- addition ∨” garded as a sort of “Boolean intertwining” that mirrors, in a remarkable fashion, the “arithmetic intertwining” between addition and multiplication in num- ber fields and local fields, which is, in some sense, the main object of study in inter-universal Teichmüller theory [cf. the discussion of Example 2.4.6, (iii); the discussion surrounding (TrHrc) in §3.10]. The above symbolic representation of the essential logical structure of inter-universal Teichmüller theory arises naturally from considerations concerning such key topics as · the coricity/symmetry/commutativity properties of the log-theta- lattice [cf. Example 3.2.2] and the closely related significance of working with both Frobenius-like and étale-like objects [cf. §3.2]; 6 SHINICHI MOCHIZUKI · the closely intertwined properties of theta functions and [p-adic/archi- medean] logarithms [cf. Examples 3.3.2, 3.3.3, 3.8.3, 3.8.4] in the con- text of the log-Kummer-correspondence [cf. §3.3; §3.11; Examples 3.2.1, 3.3.1]; · generalities concerning gluings [cf. §3.1, §3.4, §3.5, §3.10; Examples 3.1.1, 3.5.2, 3.10.1, 3.10.2]; · generalities concerning indeterminacies [cf. §3.5, §3.6, §3.7; Example 3.5.1]; · generalities concerning inter-universality [cf. §3.8; Examples 3.8.1, 3.8.2, 3.8.3, 3.8.4]; · generalities concerning descent to underlying structures [cf. §3.9, §3.10, §3.11]. Acknowledgements The present paper benefited substantially from numerous discussions with Ben- jamin Collas, Ivan Fesenko, Yuichiro Hoshi, Fumiharu Kato, Emmanuel Lepage, Arata Minamide, Wojciech Porowski, Mohamed Saı̈di, Fucheng Tan, and Shota Tsujimura. The author would like to express his appreciation to all of these math- ematicians for the time and effort that they contributed to these discussions. Section 1: Summary of non-mathematical aspects for non-specialists We begin with an overall summary of non-mathematical aspects of the situation surrounding [IUTchI-IV], which may be of interest to both non-mathematicians and mathematicians. We also refer to [FsADT], [FKvid], [FsDss], [FsPio] for a discussion of various aspects of this situation from slightly different points of view. §1.1. Publication of [IUTchI-IV] The four main papers [IUTchI-IV] on inter-universal Teichmüller theory (IUT) were accepted for publication in the Publications of the Research Institute for Mathematical Sciences (PRIMS) on February 5, 2020. This was announced at an online video news conference held at Kyoto University on April 3, 2020. The four papers were subsequently published in several special volumes of PRIMS, a leading international journal in the field of mathematics with a distinguished history dating back over half a century. The refereeing for these Special Volumes was overseen by an Editorial Board for the Special Volumes chaired by Professors Masaki Kashiwara and Akio Tamagawa. [Needless to say, as the author of these four papers, I was completely excluded from the activities of this Editorial Board for the Special Volumes.] Professor Kashiwara, a professor emeritus at RIMS, Kyoto University, is a global leader in the fields of algebraic analysis and representation theory. Professor Tamagawa, currently a professor at RIMS, Kyoto University, is a leading pioneer in the field of anabelian geometry and related research in arithmetic geometry. Here, it should be noted that, to a substantial extent, LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 7 inter-universal Teichmüller theory arose as an extension/application developed by the author in the highly mathematically stimulating en- vironment at RIMS, Kyoto University, over the course of roughly two decades [i.e., 1992 - 2012] of precisely the sort of anabelian geome- try that was pioneered by Tamagawa. It is for this reason that PRIMS stood out among mathematics journals worldwide as the most appropriate i.e., in the sense of being by far the most [and indeed perhaps the only truly] technically qualified journal for the task of refereeing and publishing the four papers [IUTchI-IV] on inter-universal Teichmüller theory. Both Professors Kashiwara and Tamagawa have an outstandingly high inter- national reputation, built up over distinguished careers that span several decades. It is entirely inconceivable that any refereeing process overseen by these mathemati- cians might be conducted relative to anything less than the highest mathematical standards, free of any inappropriate non-mathematical considerations. In an arti- cle in the Asahi Shimbun [a major Japanese newspaper] published shortly after the announcement of April 3, 2020, Professor Tamagawa is quoted as saying that he has “100 percent confidence in the refereeing” that was done for the four papers [IUTchI-IV]. In another article in the Asahi Shimbun [also published shortly after the an- nouncement of April 3, 2020], Professors Shigefumi Mori, a professor emeritus at RIMS, Kyoto University, and Nobushige Kurokawa, a professor emeritus at the Tokyo Institute of Technology, express their expectations about the possibility of applying inter-universal Teichmüller theory to other unsolved problems in number theory. Subsequent to these developments in 2020, a sequel [ExpEst] to the four original papers [IUTchI-IV] on inter-universal Teichmüller theory was accepted for publica- tion in the Kodai Mathematical Journal in September 2021. This sequel [ExpEst] concerns explicit numerical estimates in inter-universal Teichmüller theory and con- tains, in particular, a new proof of Fermat’s Last Theorem. In particular, the results proven in the four original papers [IUTchI-IV] on inter-universal Teichmüller theory, as well as the sequel [ExpEst], may now be quoted in the mathematical literature as results proven in papers that have been published in leading international journals in the field of mathematics after under- going, in the case of the four original papers [IUTchI-IV], an exceptionally thorough [seven and a half year long] refereeing process. §1.2. Redundancy assertions of the “redundant copies school” (RCS) Unfortunately, it has been brought to my attention that, despite the develop- ments discussed in §1.1, fundamental misunderstandings concerning the math- ematical content of inter-universal Teichmüller theory persist in certain sectors of the mathematical community. These misunderstandings center around a certain oversimplification which is patently flawed, i.e., leads to an immediate contradic- tion of inter-universal Teichmüller theory. This oversimplified version of inter- universal Teichmüller theory is based on assertions of redundancy concerning 8 SHINICHI MOCHIZUKI various multiple copies of certain mathematical objects that appear in inter- universal Teichmüller theory. In the present paper, I shall refer to the school of thought [i.e., in the sense of a “collection of closely interrelated ideas”] constituted by these assertions as the “RCS”, i.e., “redundant copies school [of thought]”. One fundamental reason for the use of this term “RCS” [i.e., “redundant copies school [of thought]”] in the present paper, as opposed to proper names of math- ematicians, is to emphasize the importance of concentrating on mathematical content, as opposed to non-mathematical i.e., such as social, political, or psy- chological aspects or interpretations of the situation. Thus, in a word, the central assertions of the RCS may be summarized as follows: Various multiple copies of certain mathematical objects in inter-universal Teichmüller theory are redundant and hence may be identified with one another. On the other hand, once one makes such identifications, one obtains an immediate contradiction. In the present paper, I shall refer to redundancy in the sense of the assertions of the RCS as “RCS-redundancy”, to the identifications of RCS-redundant copies that appear in the assertions of the RCS as “RCS-identifications”, and to the over- simplified version of inter-universal Teichmüller theory obtained by implementing the RCS-identifications as “RCS-IUT”. As discussed in [Rpt2018] [cf., especially, [Rpt2018], §18], there is absolutely no doubt that RCS-IUT is indeed a meaningless and absurd theory that leads immediately to a contradiction. A more technical discussion of this contradiction, in the language of inter-universal Teichmüller theory, is given in §3.1 below, while digested versions in more elemen- tary language of the technical discussion of §3 may be found in Examples 2.4.5, 2.4.7, 2.4.8, below. Rather, the fundamental misunderstandings underlying the RCS lie in the as- sertions of RCS-redundancy. The usual sense of the word “redundant” suggests that there should be some sort of equivalence, or close logical relationship, between the original version of the theory [i.e., IUT] and the theory obtained [i.e., RCS-IUT] by implementing the RCS-identifications of RCS-redundant objects. In fact, however, implementing the RCS-identifications of RCS-redundant objects radi- cally alters/invalidates the essential logical structure of IUT in such a fundamental way that it seems entirely unrealistic to verify any sort of “close logical relationship” between IUT and RCS-IUT. A more technical discussion of the three main types of RCS-redundancy/RCS- identification which we refer to as “(RC-FrÉt)”, “(RC-log)”, and “(RC-Θ)” is given, in the language of inter-universal Teichmüller theory, in §3.2, §3.3, §3.4, LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 9 below. In fact, however, the essential mathematical content of these three main types of RCS-redundancy/RCS-identification is entirely elementary and lies well within the framework of undergraduate-level mathematics. A discussion of this essentially elementary mathematical content is given in §2.3, §2.4 below [cf., especially, Examples 2.4.5, 2.4.7, 2.4.8]. One important consequence of the technical considerations discussed in §3 below is the following: from the point of view of the logical relationships between various as- sertions of the RCS, the most fundamental type of RCS-redundancy is (RC-Θ). That is to say, (RC-Θ) may be understood as the logical cornerstone of the various assertions of the RCS. §1.3. Qualitative assessment of assertions of the RCS As discussed in detail in §3.4 below [cf. also §2.3, §2.4], implementing the logical cornerstone RCS-identification of (RC-Θ) completely invalidates the crucial logical AND “∧” property satis- fied by the Θ-link a property that underlies the entire logical struc- ture of inter-universal Teichmüller theory. In particular, understanding the issue of how the RCS treats this fundamental conflict between the RCS-identification of (RC-Θ) and the crucial ∧-property of the Θ-link is central to the issue of assessing the assertions of the RCS. In March 2018, discussions were held at RIMS with two adherents of the RCS concerning, in particular, (RC-Θ) [cf. [Rpt2018], [Dsc2018]]. Subsequent to these discussions, after a few e-mail exchanges, these two adherents of the RCS informed me via e-mail in August 2018 in response to an e-mail that I sent to them in which I stated that I was prepared to continue discussing inter-universal Teichmüller the- ory with them, but that I had gotten the impression that they were not interested in continuing these discussions that indeed they were not interested in continuing these discussions concerning inter-universal Teichmüller theory. In the same e-mail, I also stated that perhaps it might be more productive to continue these discussions of inter-universal Teichmüller theory via different participants [i.e., via “represen- tatives” of the two sides] and encouraged them to suggest possible candidates for doing this, but they never responded to this portion of my e-mail. [Incidentally, it should be understood that I have no objection to making these e-mail messages public, but will refrain from doing so in the absence of explicit permission from the two recipients of the e-mails.] Since March 2018, I have spent a tremendous amount of time discussing the fundamental “(RC-Θ) vs. ∧-property” conflict mentioned above with quite a number of mathematicians. Moreover, during the years following the March 2018 discussions, many mathematicians [including myself!] with whom I have been in contact have devoted a quite substantial amount of time and effort to analyzing and discussing certain 10pp. manuscripts written by adherents of the RCS [cf., especially, the discussion of the final page and a half of the files “[SS2018-05]”, 10 SHINICHI MOCHIZUKI “[SS2018-08]” available at the website [Dsc2018]] indeed to such an extent that by now, many of us can cite numerous key passages in these manuscripts by memory. More recently, one mathematician with whom I have been in contact has made a quite intensive study of the mathematical content of recent blog posts by adherents of the RCS. Despite all of these efforts, the only justification for the logical cornerstone RCS-identification of (RC-Θ) that we [i.e., I myself, together with the many mathematicians with whom I have discussed these issues] could find either in oral explanations during the discussions of March 2018 or in subsequent written records produced by adherents of the RCS [i.e., such as the 10pp. manuscripts referred to above or various blog posts] were statements of the form “I don’t see why not”. [I continue to find it utterly bizarre that such justifications of the assertions of the RCS appear to be taken seriously by some professional mathematicians.] In particular, we were unable to find any detailed mathematical discussion by adherents of the RCS of the fundamental “(RC-Θ) vs. ∧-property” conflict mentioned above. That is to say, in summary, the mathematical justification for the “redundancy” asserted in the logical cornerstone assertion (RC-Θ) of the RCS remains a complete mystery to myself, as well as to all of the mathematicians that I have consulted concerning this issue [cf. the discussion of Examples 2.4.5, 2.4.7, 2.4.8]. Put another way, the response of all of the mathematicians with whom I have had technically meaningful discussions concerning the assertions of the RCS was completely uniform and unanimous, i.e., to the effect that these assertions of the RCS were obviously completely math- ematically inaccurate/absurd, and that they had no idea why adherents of the RCS continued to make such manifestly absurd assertions. In particular, it should be emphasized that I continue to search for a professional mathematician [say, in the field of arithmetic geometry] who feels that he/she understands the mathematical content of the assertions of the RCS and is willing to discuss this math- ematical content with me or other mathematicians with whom I am in contact [cf. the text at the beginning of [Dsc2018]]. It is worth noting that this situation also constitutes a serious violation of article (6.) Mathematicians should not make public claims of potential new theorems or the resolution of particular mathematical problems unless they are able to provide full details in a timely manner. of the subsection entitled “Responsibilities of authors” of the Code of Practice of the European Mathematical Society (cf. [EMSCOP]). In this context, one important observation that should be kept in mind is the following [cf. the discussion of [Rpt2018], §18]: LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 11 (UndIg) There is a fundamental difference between (UndIg1) criticism of a mathematical theory that is based on a solid, technically accurate understanding of the content and logi- cal structure of the theory and (UndIg2) criticism of a mathematical theory that is based on a funda- mental ignorance of the content and logical structure of the theory. An elementary classical example of this sort of difference is discussed in §2.1 below. In the case of the RCS, the lack of any thorough mathematical discussion of the fundamental “(RC-Θ) vs. ∧-property” conflict mentioned above in the vari- ous oral/written explanations set forth by adherents of the RCS demonstrates, in a definitive way, that none of the adherents of the RCS has a solid, technically accu- rate understanding of the logical structure of inter-universal Teichmüller theory in its original form, i.e., in particular, of the central role played in this logical structure by the “∧-property” of the Θ-link. Put another way, the only logically consistent explanation of this state of affairs is that the theory “RCS-IUT” that adherents of the RCS have in mind, i.e., the theory that is the object of their criticism, is simply a completely different and logically unrelated theory from the theory constituted by inter-universal Teichmüller theory in its original form. Finally, it should be mentioned that although some people have asserted par- allels between the assertions of the RCS and the fundamental error in the first version of Wiles’s proof of the Modularity Conjecture in the mid-1990’s, this anal- ogy is entirely inappropriate for numerous reasons. Indeed, as is well-known, nothing even remotely close to the phenomena discussed thus far in the present §1.3 occurred in the case of the error in the first version of Wiles’s proof. The fact that there was indeed a fatal error in the first version of Wiles’s proof was never disputed in any way by any of the parties involved; the only issue that arose was the issue of whether or not the proof could be fixed. By contrast, no essential er- rors have been found in inter-universal Teichmüller theory, since the four preprints [IUTchI-IV] on inter-universal Teichmüller theory were released in August 2012. That is to say, in a word, the assertions of the RCS are nothing more than mean- ingless, superficial misunderstandings of inter-universal Teichmüller theory on the part of people who are clearly not operating on the basis of a solid, technically accurate understanding of the mathematical content and essential logical structure of inter-universal Teichmüller theory. §1.4. The importance of extensive, long-term interaction In general, the transmission of mathematical ideas between individuals who share a sufficient stock of common mathematical culture may be achieved in a relatively efficient way and in a relatively brief amount of time. Typical ex- amples of this sort of situation in the context of interaction between professional mathematicians include · one-hour mathematical lectures, · week-long mathematical lecture series, and · informal mathematical discussions for several days to a week. 12 SHINICHI MOCHIZUKI In the context of mathematical education, typical examples include · written or oral mathematical examinations and · mathematics competitions. The successful operation of each of these examples relies, in an essential way, on a common framework of mathematical culture that is shared by the various participants in the activity under consideration. On the other hand, in the case of a fundamentally new area of research, such as inter-universal Teichmüller theory, which evolved out of research over the past quarter of a century concerning absolute anabelian geometry, certain types of categories arising from arithmetic geometry, and certain arithmetic aspects of theta functions, the collection of mathematicians who share such a sufficient stock of common mathematical culture tends to be relatively small in number. In par- ticular, for most mathematicians even many arithmetic geometers or anabelian geometers short-term interaction of the sort that occurs in the various typical examples mentioned above is far from sufficient to achieve an effective trans- mission of mathematical ideas. That is to say, no matter how mathematically talented the participants in such platforms of interaction may be, it takes time for the participants to · analyze and sort out numerous mutual misunderstandings, · develop effective techniques of communication that can transcend such misunderstandings, and · digest and absorb new ideas and modes of thought. Depending on the mathematical content under consideration, as well as on the mathematical talent, mathematical background, and time constraints of the partic- ipants, this painstaking process of analysis/development/digestion/absorption may require patiently sustained efforts to continue constructive, orderly mathematical discussions [via e-mail, online video discussions, or face-to-face meetings] over a period of months or even years to reach fruition. Indeed, my experience in exposing the ideas of inter-universal Teichmüller theory to numerous mathematicians over the past decade suggests strongly that, in the case of inter-universal Teichmüller theory, it is difficult to expedite this process to the extent that it can be satisfactorily achieved in less than half a year or so. In particular, in the case of inter-universal Teichmüller theory, a week-long session of discussions such as the discussions held at RIMS in March 2018 with two adherents of the RCS [cf. [Rpt2018], [Dsc2018]] is far from sufficient. This is something that I emphasized, both orally during these discussions and in e-mails to these two adherents of the RCS during the summer of 2018 subsequent to these discussions. §1.5. The historical significance of detailed, explicit, accessible records As was discussed in §1.3, I continue to search for a professional mathematician [say, in the field of arithmetic geometry] who purports to understand the math- LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 13 ematical justification for the RCS-redundancy asserted in the logical corner- stone assertion (RC-Θ) i.e., in particular, who has confronted the mathematical content of the fundamental “(RC-Θ) vs. ∧-property” conflict mentioned in §1.3 and who is prepared to discuss this mathematical content with me or other mathematicians with whom I am in contact. Of course, a detailed, explicit, mathematically substantive, and readily ac- cessible written exposition i.e., as an alternative to direct mathematical discussions [via e-mail, online video discussions, or face-to-face meetings] of the mathematical justification for the log- ical cornerstone assertion (RC-Θ) would also be quite welcome [cf. the discussion of [Rpt2014], (7)]. Moreover, in this context, it should be emphasized that such a detailed, explicit, mathematically substantive, and readily accessible written expo- sition would be of great value not only for professional mathematicians and graduate students who are involved with inter-universal Teichmüller theory at the present time, but also for scholars in the [perhaps distant!] future. In general, it cannot be overemphasized that maintaining such detailed, ex- plicit, mathematically substantive, and readily accessible written records is of fundamental importance to the development of mathematics. Indeed, as was discussed in the final portion of [Rpt2018], §3, from a historical point of view, it is only by maintaining such written records that the field of math- ematics can avoid the sort of well-known and well-documented confusion that lasted for so many centuries concerning “Fermat’s Last Theorem”. Moreover, it is fascinating to re-examine, from the point of view of a modern observer, the in- tense debates that occurred, during the time of Galileo, concerning the theory of heliocentrism or, during the time of Einstein, concerning the theory of relativity. Perhaps a more pertinent example [as was pointed out to the author by Fu- miharu Kato], relative to the issue of “redundant copies”, may be found in the well-known dispute i.e., concerning “algebraic truths” [which corresponds to Weierstrass’ approach] versus “geometric fantasies” [which corresponds to Rie- mann’s approach] in the 19-th century between Weierstrass and Riemann con- cerning approaches to complex function theory [cf. [Btt]]. That is to say, the criticism by Weierstrass of the “geometric fantasies” approach due to Riemann via analytic continuation on Riemann surfaces, which are obtained by gluing to- gether what are, perhaps to some observers, seemingly “redundant”! copies of open subsets of the complex plane [cf. the discussion of the well-known gluing construction of the projective line in Example 2.4.7 below; the discussion of ana- lytic continuation of the complex logarithm in the discussion surrounding (FxEuc) in §3.1 below, as well as in Example 3.10.1 below; the illustrations of [AnCnCv], [AnCnLg]!], exhibits remarkable parallels in spirit to the assertions of the RCS. Here, it is also of interest to note [as was pointed out to the author by Ben- jamin Collas] that the issue of analytic continuation of the complex logarithm is very closely related to the long and heated dispute between Leibniz and [Johann] Bernoulli concerning the appropriate definition of logarithms of negative and more general complex numbers a dispute that was ultimately, in some sense, resolved by Euler’s formula and the acceptance of the multi-valued nature of the complex 14 SHINICHI MOCHIZUKI logarithm [cf. the discussion of “Euler’s Formula and Its Consequences” in [AnHst], Chapter I, §I.5, although it must be kept in mind that in Euler’s time, complex function theory, i.e., of the sort necessary to treat the complex logarithm as a holomorphic function, had not yet been developed]. This multi-valued nature of the complex logarithm is closely related not only to the theory of analytic continuation of the complex logarithm, but also to the theory of coverings [in the sense of the classical topological fundamental group], where the indeterminacy in values may be understood as an indeterminacy in the choice of basepoint, that is to say, as a sort of distant though nonetheless quite direct! ancestor of the inter-universal indeterminacies that appear in inter-universal Teichmüller theory [cf. the discussion of §3.8, especially Example 3.8.1, below!]. The central role occupied by the notion of analytic continuation in these his- torical disputes of Weierstrass/Riemann and Leibniz/Bernoulli is also of interest, in the context of inter-universal Teichmüller theory, in light of the central role played, in the history of analytic continuation, by the functional equation [i.e., Jacobi identity] for the theta function, which exhibits numerous remarkable structural similarities to inter-universal Teichmüller theory [cf. the discussion of Example 3.3.2 below], and which is also closely related to Riemann’s famous research con- cerning the theory of the functional equation/analytic continuation of the Riemann zeta function. Research in the 19-th century concerning analytic continuation may also be regarded as a sort of early precursor of more modern notions such as monodromy representations and connections/crystals an observation which is of interest in light of the analogy between connections/crystals and the notion of multiradiality in inter-universal Teichmüller theory [cf. the discussion of [Alien], §3.1, (v), as well as the discussion of §3.5, §3.10 below]. Before proceeding, we remark that it is interesting to observe that this his- torical discussion of the functional equation of the theta function in the context of analytic continuation, in this case on the upper half-plane, is reminiscent of the famous observation of Poincaré i.e., in the form of an idea that came to him while in transit during his travels that “the transformations” that he had used “to define Fuchsian functions were identical with those of non-Euclidian geometry” [cf. [Pnc], p. 53]. This famous observation concerning the isomorphic nature of the group of transformations of a modular function i.e., such as the theta function and a certain group of symmetries of the [“non-Euclidean”] hyperbolic geometry of the upper half-plane seems to be one of the principal motivations behind the famous quote, due to Poincaré, that “mathematics is the art of giving the same name to different things”, i.e., “things which differ in matter, but are similar in form” [cf. the discussion of [Pnc], pp. 34 35]. Moreover, this remarkable train of thought, which was recorded for posterity by Poincaré in [Pnc], is of particular interest in the context of the present discussion of the relationship between various classical notions and inter-universal Teichmüller theory in that it seems almost prescient in its deep resemblance to the main notions namely, coricity and multiradiality that underlie the concept of inter-universality in inter-universal Teichmüller theory. Indeed: (SmDff1) The search for coric structures in inter-universal Teichmüller theory may be thought of precisely as the search for the “same name” for suitable LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 15 portions of structures on opposite sides i.e., “different things” of the Θ- or log-links in inter-universal Teichmüller theory. Moreover, perhaps the most fundamental instance of such a coric structure in inter-universal Teichmüller theory consists of the abstract topological groups, regarded up to isomorphism, underlying the Galois groups/arithmetic fundamental groups that arise from rings/schemes on opposite sides of the Θ- or log- links. (SmDff2) The search for multiradial structures in inter-universal Teichmüller the- ory may be thought of precisely as the search for the “same name” i.e., more precisely, in the form of an isomorphism up to suitable indetermina- cies for · a suitable portion of the system [or portion of the log-theta- lattice] consisting of distinct ring/scheme structures linked by the Θ-link, on the one hand, and · a suitable portion of a single ring/scheme structure, on the other i.e., for “things which differ in matter”. More technical details may be found in the discussion of coricity and inter-universality in §3.2, §3.8, below, as well in the discussion of descent via the multiradial algorithms of inter-universal Teichmüller theory given in §3.10, §3.11, below. Of course, from the point of view of a modern observer who is well-versed in axiomatic set theory and the theory of [Grothendieck] universes, the bitter historical disputes of Weierstrass/Riemann and Leibniz/Bernoulli discussed above seem somewhat quaint or even “silly”. One should not, however, in this context, overlook the importance of such bitter historical disputes in motivating the develop- ment of such modern tools as axiomatic set theory and the theory of [Grothendieck] universes, which underlie this privileged viewpoint of the modern observer. Con- versely, when viewed from this historical perspective, the assertions of the RCS seem all the more like a sort of bizarre anachronism, which has no place in the 21-st century [cf., especially, the entirely elementary nature of the various examples such as the gluing construction of the projective line discussed in Example 2.4.7 that appear in §2 below]! Moreover, in the case of the Leibniz/Bernoulli dispute, it is of interest to note especially, in the context of the highly sensationalist nature of the coverage of inter-universal Teichmüller theory in the English-language mass-media and internet [cf., e.g., the discussion of §1.8, §1.10, §1.12 below] that apparently, at the time of the dispute, the dispute was kept as secret as possible in order to avoid “damaging the prestige of pure mathematics as an exact and rigorous science” [cf. the discussion of “Euler’s Formula and Its Consequences” in [AnHst], Chapter I, §I.5]. At any rate, in the context of this sort of historical discussion, it cannot be overemphasized that all of these historical re-examinations [i.e., of the sort that underlie the discussion of the last few paragraphs!] are technically possible precisely because of the existence of detailed, explicit, mathematically sub- stantive, and readily accessible written expositions of the logical structure underlying the various central assertions that arose in the de- bate. 16 SHINICHI MOCHIZUKI In this context, it should be noted that, from a historical point of view, one pattern that typically underlies the formidable deadlocks that tend to occur in such debates is the point of view, on the part of parties opposed to a newly developed theory, that (CmSn) it is a “matter of course” or “common sense” i.e., in the lan- guage of the above discussion, a matter that is so profoundly self-evident that any “decent, reasonable observer” would undoubtedly find detailed, explicit, mathematically substantive, and readily accessible written expo- sitions of its logical structure to be entirely unnecessary that the issues under consideration can be completely resolved within some exist- ing, familiar framework of thought without the introduction of the newly developed theory, which is regarded as deeply disturbing and unlikely to be of use in any substantive mathematical sense cf., e.g., the discussion of Example 1.5.1 below. In fact, however, (OvDlk) ultimately, the only meaningful technical tool that humanity can apply to develop the cultural infrastructure necessary to overcome such deadlocks is precisely the production of detailed, explicit, mathematically substantive, and readily accessible written expositions of the logical structure underlying the points of view that are regarded by certain parties as being a “matter of common sense” [cf. the discussion of [EMSCOP] in §1.3; the discussion of (UndIg) in §1.3; the discussion of §1.10 below; §2.1 below; the discussion of (FxRng), (FxEuc), (FxFld), (RdVar) in §3.1 below]. Example 1.5.1: Irrationality, impartiality, and the Voodoo Hypothesis. (i) We begin by recalling the very classical and elementary proof that, for any prime number p, p is irrational, i.e., that there does not exist any rational number whose square is equal to p. Indeed, if there did exist a rational number r = a/b, where a and b are relatively prime nonzero integers, such that r 2 = p, then the resulting relation a 2 = p · b 2 would violate the unique factorization property satisfied by the nonzero integers. This unique factorization property may, in turn, by verified by applying the Euclidean division algorithm. (ii) The discovery of the irrationality of square roots of prime numbers dis- cussed in (i) is typically attributed, in the case p = 2 [in which case the Euclidean division algorithm amounts, in essence, to the classification of integers into odd and even integers], to the ancient Greek philosopher Hippasus. Apparently, this discovery arose in the context of applying the Pythagorean Theorem [cf. the discus- sion of Examples 1.10.1, 1.12.1 below] to compute the length of the diagonal of a square with sides of length 1. The Pythagorean school was reported to have found this discovery to be shocking and deeply disturbing, and indeed it seems that this negative appraisal of Hippasus’ discovery may have led to the death of Hippasus by drowning. (iii) The sort of negative appraisal that occurred in (ii) could easily arise in the mind of any observer indeed, even modern observers [such as students] who are not familiar with the sort of abstract mathematical reasoning that underlies the LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 17 proof of (i) who has a deep sense of confidence in his/her understanding of the “common sense definition of mathematics” as the study of explicit computations involving rational numbers, e.g., of the sort that may be done with a desktop calculator [or, in earlier centuries, with an abacus!]. [We refer to Example 1.5.2 below for further similar examples of this sort of phenomenon.] Such a negative appraisal on the part of a strongly computationally oriented observer might also be additionally supported by (HeurBlf) a heuristically supported belief that “surely” if one is given sufficient time and manpower to perform suitable computations to sufficiently high order, then there is no doubt that one should be able to find a rational square root of 2. Moreover, depending on the social/political circumstances surrounding the situa- tion, even third parties who are ignorant of the details of situation might be led (FsObjImpl) to [falsely!] assert that the only truly objective or impartial way to treat the two schools i.e., of people who assert the irrationality of 2 via the proof of (i) and people who doubt this argument is to talk as if the issue of the irrationality of 2 is unresolvable or even to refuse to discuss the issue at all. Indeed, it is worth recalling that ancient accounts of Hippasus and the Pythagorean school are a stark reminder of just how dire the consequences can be when those who are socially/politically recognized as “objective arbiters” for a mathematical dispute act on the basis of a grossly mathematically inaccurate understanding of the situation. (iv) In the context of the discussion of (iii), it is useful to refer to a situation [i.e., such as the situation discussed in (iii)] in which the validity of a mathematical proof is called into question, not on the basis of some sort of logical defect [i.e., such as a gap in the proof], but on the basis of some sort of heuristically based belief [cf. (HeurBlf)] to the effect that “surely” if one is given sufficient time and manpower to sort through various technical details, then there is no doubt that one should be able to find some sort of substantive problem with the proof such as a counterexample or fallacious reasoning, i.e., some sort of “voodoo” as an invocation of the “Voodoo Hypothesis”. Once the Voodoo Hypothesis has been invoked, mathematicians who have a mathematically accurate, rigorous understanding of the proof in question are then often portrayed as being nothing more than mindlessly obedient zombies, i.e., who are acting solely or essentially under the influence of the “voodoo” applied in the proof. Indeed, this is precisely the sort of situation that has developed, in certain sectors of the mathematical community, concerning inter-universal Teichmüller theory cf. the discussion of RCS-IUT in §1.2, §1.3, as well as §1.8, §1.10 below; Examples 2.4.5, 2.4.7, 2.4.8 below; the discussion of (FxRng), (FxEuc), (FxFld), (RdVar) in §3.1 below. (v) At this point, we observe that the discussion of (ii), (iii), and (iv) prompt the following question: What lessons may be learned from the discussion in (ii), (iii), and (iv) of the negative appraisal of the proof of the irrationality of 2? 18 SHINICHI MOCHIZUKI First of all, it is important to remember that (Blf =Pf) heuristically based beliefs, as in (HeurBlf), are not mathematical proofs and, in particular, can never serve as a “viable substitute” for a rigorous mathematical proof. Secondly, it is important to remember [cf. (FsObjImpl)] that (MthAcc) a truly objective/impartial position concerning a mathematical dis- pute can only be achieved as a result of a mathematically accurate understanding of the mathematics involved and, in particular, can never be achieved through an ignorance of the mathematics involved. In this context, it is interesting to note that (MthAcc) has important consequences from the point of view of the topic of computer verification of mathematical assertions [cf. the discussion surrounding (CmpVer) in §1.12 below]. That is to say, for instance, in the case of the proof of (i), although it is not so technically difficult to formalize this [relatively simple!] argument in such a way that the argument could be “verified” by a computer, (SocPol) such a computer verification becomes entirely socially/politically meaningless in situations in which parties such as the computation- ally oriented observer of (iii)! who do not share or recognize the abstract conceptual mathematical framework that necessarily underlies any sort of computer-ready formalization of the proof of (i) hold a monopoly on so- cial/political authority. Thus, in summary, from a historical point of view, it seems that the main lesson to be learned from the situation discussed in (ii), (iii), and (iv) is (LTInfr) the fundamental importance in the context of dealing in a mean- ingful and effective manner with mathematical disputes of maintain- ing a long-term infrastructural apparatus for directly confronting, disseminating, and further developing the mathematics involved [cf. the discussion surrounding (BlkAcc) in §1.12 below]. Needless to say, the starting point of the activities of such a long-term infras- tructural apparatus necessarily lies in the production of detailed, explicit, mathe- matically substantive, and readily accessible written expositions of the logical structure underlying the points of view that are regarded by various parties as being a “matter of common sense”. Example 1.5.2: The internet/mass media as an apple of discord. In Ex- ample 1.5.1 [cf., especially, the discussion of “modern observers” in Example 1.5.1, (iii)], the particular example of the issue of the irrationality of 2 was examined in detail. In fact, however, this sort of situation exists in quite substantial abundance throughout mathematics and especially throughout the sort of elementary mathe- matics that is commonly covered in primary and secondary, as well as in university, eduation. Well-known examples of this phenomenon include the following: (NegInt) the sense, in the context of multiplication of positive and negative inte- gers, that any product of two negative integers must be a negative integer, i.e., on the grounds that the appearance of “two minus signs” in the prod- uct “surely” results in output that is “all the more negative!”; LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 19 (MxDiag) the sense, in the context of elementary linear algebra, that “surely” all matrices are diagonalizable, i.e., if one just tries hard enough to find the “right basis!”. Moreover, unlike the case with the ancient context discussed in Example 1.5.1, (ii), this sort of phenomenon can be substantially further exacerbated in modern con- texts by the internet/mass media, which exhibits a conspicuous tendency to operate as a sort of “apple of discord” that has the effect of not only creating, but also amplifying to often absurd proportions, an artificial socio-political dynamic that fuels a burning desire to leap to conclusions and achieve “instant satisfaction” with respect to some sort of essentially meaningless fictional/delusional mirage, e.g., with respect to achieving an understanding of some sort of apparently puzzling mathe- matical phenomenon of the sort described in (NegInt), (MxDiag) that appears, to some observers, to defy their deep heuristic “common sense” understanding of the situation cf. the discussion of §1.8, §1.10, §1.12, below. Finally, we observe that, just as in the case of the discussion surrounding (LTInfr) in Example 1.5.1, (v), the only way, to the knowledge of the author at the time of writing, to overcome the detrimental effects of such artificial socio-political dynamics lies in maintaining a long-term infrastructural apparatus for directly confronting, disseminating, and further developing the mathematics involved, the starting point of which inevitably involves the production of detailed, explicit, mathematically substantive, and readily accessible written expositions of the logical structure underlying the points of view that are regarded by various parties as being a “matter of common sense”. §1.6. The importance of further dissemination One fundamental and frequently discussed theme in the further development of inter-universal Teichmüller theory is the issue of increasing the number of profes- sional mathematicians who have a solid, technically accurate understanding of the details of inter-universal Teichmüller theory. Indeed, this issue is in some sense the central topic of [Rpt2013], [Rpt2014]. As discussed in §1.4, in order to achieve such a solid, technically accurate understanding of the theory, it is neces- sary to devote a substantial amount of time and effort over a period of roughly half a year to two or three years, depending on various factors. It also requires the participation of professional mathematicians or graduate students who are · sufficiently familiar with numerous more classical theories in arithmetic geometry [cf. the discussion of [Alien], §4.1, (ii); [Alien], §4.4, (ii)], · sufficiently well motivated and enthusiastic about studying inter-universal Teichmüller theory, and · sufficiently mathematically talented, and who have a · sufficient amount of time to devote to studying the theory. As a result of quite substantial dissemination efforts not only on my part, but also on the part of many other mathematicians, the number of professional mathe- maticians who have achieved a sufficiently detailed understanding of inter-universal Teichmüller theory to make independent, well informed, definitive statements con- cerning the theory that may be confirmed by existing experts on the theory [cf. also the discussion of §1.7 below] is roughly on the order of 10. It is worth noting that 20 SHINICHI MOCHIZUKI although this collection of mathematicians is centered around RIMS, Kyoto Uni- versity, it includes mathematicians of many nationalities and of age ranging from around 30 to around 60. One recent example demonstrated quite dramatically that it is quite possible to achieve a solid mathematical understanding of inter-universal Teichmüller theory as a graduate student by studying on one’s own, outside of Japan, and with essentially zero contact with RIMS, except for a very brief pe- riod of a few months at the final stage of the student’s study of inter-universal Teichmüller theory. Finally, we observe, in the context of the discussion [cf. §1.3, §1.4, §1.5] of the assertions of the RCS, that another point that should be emphasized is that it is also of fundamental importance to increase the number of professional mathematicians [say, in the field of arithmetic geometry] who have a solid technical understanding of the mathematical content of the assertions of the RCS, and who are pre- pared to discuss this mathematical content with members of the “IUT community” [i.e., with mathematicians who are substantially involved in mathematical research and/or dissemination activities concerning inter-universal Teichmüller theory]. Here, we note in passing that such a solid technical understanding of the mathematical content of the assertions of the RCS is by no means “equivalent” to expressions of support for the RCS on the basis of non-mathematical i.e., such as social, political, or psychological reasons. In this context, it should also be emphasized and understood [cf. the discussion of [Rpt2014], (7)] that both · producing detailed, explicit, mathematically substantive, and readily ac- cessible written expositions of the mathematical justification of assertions of the RCS [such as (RC-Θ)!], i.e., as discussed in §1.5, and · increasing the number of professional mathematicians [say, in the field of arithmetic geometry] who have a solid technical understanding of the mathematical content of the assertions of the RCS, and who are prepared to discuss this mathematical content with members of the IUT community are in the interest not only of the IUT community, but of the RCS as well. More- over, the process of attaining a solid, technically accurate understanding of the precise logical relationship between RCS-IUT and IUT, i.e., as ex- posed, for instance, in the present paper, can serve as a valuable peda- gogical tool [cf. the discussion of [Rpt2018], §17] for mathematicians currently in the process of studying inter-universal Teichmüller theory. §1.7. The notion of an “expert” One topic that sometimes arises in the context of discussions of dissemination of inter-universal Teichmüller theory [i.e., as in §1.6], is the following issue: What is the definition of, or criterion for, being an “expert” on inter- universal Teichmüller theory? LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 21 In a word, it is very difficult to give a brief, definitive answer, e.g., in the form of a straightforward, easily applicable criterion, to this question. On the other hand, in this context, it should also be pointed out that the difficulties that arise in the case of inter-universal Teichmüller theory are, in fact, not so qualitatively different from the difficulties that arise in answering the analogous question for mathematical theories other than inter-universal Teichmüller theory. These difficulties arise throughout the daily life of professional mathematicians in numerous contexts, such as the following: (Ev1) preparing suitable exercises or examination problems to educate and evaluate students, (Ev2) evaluating junior mathematicians, (Ev3) refereeing/evaluating mathematical papers for journals. From my point of view, as the author of [IUTchI-IV], one fundamental criterion that I always keep in mind not only the in case of [IUTchI-IV], but also in the case of other papers that I have written, as well as when I am involved in the various types of evaluation procedures (Ev1) (Ev3) discussed above is the issue of the extent to which the level of understanding of the mathematician in question enables the mathematician to “stand on his/her own two feet” with regard to various assertions concerning the theory, on the basis of independent, logical reasoning, without needing to be “propped up” or cor- rected by me or other known experts in the theory. I often refer to this criterion as the criterion of autonomy of understanding. Of course, from a strictly rigorous point of view, this criterion is, in some sense, not so “well-defined” and, in many contexts, difficult to apply in a straightforward fashion. On the other hand, in the past, various mathematicians involved with inter-universal Teichmüller theory have demonstrated such an autonomous level of understanding in the following ways: (Atm1) the ability to detect various minor errors/oversights in [IUTchI-III]; (Atm2) the ability to propose new, insightful ways of thinking about various aspects of inter-universal Teichmüller theory; (Atm3) the ability to propose ways of modifying inter-universal Teichmüller theory so as to yield stronger or more efficient versions of the theory; (Atm4) the ability to produce technically accurate oral or written expositions of inter-universal Teichmüller theory; (Atm5) the ability to supervise or direct new mathematicians i.e., by train- ing/educating professional mathematicians or graduate students with re- gard to inter-universal Teichmüller theory who, in due time, demon- strate various of the four types of ability (Atm1) (Atm4) discussed above. Of course, just as in the case of other mathematical theories, different experts demonstrate their expertise in different ways. That is to say, experts in inter- universal Teichmüller theory often demonstrate their expertise with respect to some of these five types of ability (Atm1) (Atm5), but not others. 22 SHINICHI MOCHIZUKI In this context, it should be pointed out that one aspect of inter-universal Teichmüller theory that is currently still under development is the analogue for inter-universal Teichmüller theory of (Ev1), i.e., preparing suitable exercises for mathematicians currently in the process of studying inter-universal Teichmüller theory. This point of view may be seen in the discussion in the final portion of the Introduction to [Alien], as well as in the discussion of “valuable pedagogical tools” in [Rpt2018], §17. Indeed, many of the technical issues discussed in [Rpt2018], §15 [or, alternatively, Example 3.2.2 of the present paper], may easily be reformu- lated as “exercises” or, alternatively, as “examination problems” for evaluating the level of understanding of mathematicians in the process of studying inter-universal Teichmüller theory. §1.8. Fabricated versions spawn fabricated dramas As discussed in §1.6, §1.7, by now there is a substantial number of mathe- maticians who have attained a thorough, accurate, and automous understanding of inter-universal Teichmüller theory. In each of the cases of such mathematicians that I have observed thus far, such an understanding of the theory was achieved essen- tially by means of a thorough study of the original papers [IUTchI-IV], followed by a period of constructive discussions with one or more existing experts that typically lasted roughly from two to six months to sort out and resolve various “bugs” in the mathematician’s understanding of the theory that arose when the mathematician studied the original papers on his/her own [cf. the discussion of §1.4]. On the other hand, there is also a growing collection of mathematicians who have a somewhat inaccurate and incomplete and indeed often quite superficial understanding of certain aspects of the theory. This in and of itself is not problematic that is to say, so long as the mathematician in question maintains an appropriate level of self-awareness of the inaccurate and incomplete nature of his/her level of understanding of the theory and indeed is a phenomenon that often occurs as abstract mathematical theories are disseminated. Unfortunately, however, a certain portion of this collection of mathematicians [i.e., whose understanding of the theory is inaccurate and incomplete] have exhibited a tendency to assert/justify the validity of their inaccurate and incomplete understand- ing of the theory by means of “reformulations” or “simplifications” of the theory, which are in fact substantively different from and have no directly logical relationship to [e.g., are by no means “equivalent” to!] the original theory. Indeed, the version, referred to in the present paper as “RCS-IUT” [cf. §1.2], that arises from implementing the assertions of the RCS appears to be the most famous of these fabricated versions of inter-universal Teichmüller theory [cf. also the discussion of Example 2.4.5 below for a more detailed discussion of various closely related variants of RCS-IUT]. On the other hand, other, less famous fabricated versions of inter-universal Teichmüller theory have also come to my attention in recent years. Here, before proceeding, we note that, in general, reformulations or simplifica- tions of a mathematical theory are not necessarily problematic, i.e., so long as they LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 23 are indeed based on a thorough and accurate understanding of the original theory and, moreover, can be shown to have a direct logical relationship to the original theory. The authoring of fabricated versions of inter-universal Teichmüller theory ap- pears to be motivated, to a substantial extent, by a deep desire to recast inter- universal Teichmüller theory in a “simplified” form that is much closer to the sort of mathematics with which the author of the fabricated version is already famil- iar/feels comfortable. On the other hand, this phenomenon of producing fabricated versions also appears to have been substantially fueled by numerous grotesquely distorted mass media reports and comments on the English-language internet that blithely paint inter- universal Teichmüller theory as a sort of cult religion, fanatical political movement, mystical philosophy, or vague sketch/proposal for a mathemat- ical theory. Moreover, another unfortunate tendency, of which RCS-IUT is perhaps the most egregious example, is for fabricated versions of inter-universal Teichmüller theory to spawn lurid social/political dramas revolving around the content of the fabricated version, which in fact have essentially nothing to do with the content of inter-universal Teichmüller theory. Such lurid dramas then spawn further grotesquely distorted mass media reports and comments on the English-language internet, which then reinforce and enhance the social/political status of the fabricated version [cf. the discussion of Example 1.5.2]. Here, it should be emphasized that such vicious spirals have little [or nothing] to do with substantive mathematical content and indeed serve only to mass-produce unnecessary confusion that is entirely counterproductive, from the point of the view of charting a sound, sustainable course in the future development of the field of mathematics [cf. the discussion of [Alien], §4.4, (iv)]. In fact, of course, inter-universal Teichmüller theory is neither a religion, nor a political movement, nor a mystical philosophy, nor a vague sketch/proposal for a mathematical theory. Rather, it should be emphasized that inter-universal Teichmüller theory is a rigorously formulated mathematical theory that has been verified countless times by quite a number of mathe- maticians, has undergone an exceptionally thorough seven and a half year long refereeing process, and was subsequently published in a leading inter- national journal in the field of mathematics. [cf. the discussion of §1.1]. In particular, in order to avoid the sort of vicious spirals referred to above, it is of the utmost importance to concentrate, in discussions of inter-universal Teichmüller theory, on substantive mathematical content, as opposed to non-mathematical such as social, political, or psychological aspects or interpretations of the situation. As discussed in §1.2, this is the main reason for the use of the term “RCS” in the present paper. 24 SHINICHI MOCHIZUKI §1.9. Geographical vs. mathematical proximity Historically, mathematical interaction between professional mathematicians re- lied on physical meetings or the exchange of hardcopy documents. Increasingly, however, advances in information technology have made it possible for mathemati- cal interaction between professional mathematicians to be conducted electronically, by means of e-mail or online video communication. Of course, this does not imply that physical meetings or the exchange of hardcopy documents especially in sit- uations where physical meetings or the exchange of hardcopy documents do indeed function in a meaningful way, from the point of view of those involved should necessarily be eschewed. On the other hand, physical meetings between participants who live in dis- tant regions requires travel. Moreover, travel, depending on the situations of the participants, can be a highly taxing enterprise. Indeed, travel, as well as lodging accommodations, typically requires the expenditure of a quite substantial amount of money, as well as physical and mental effort on the part of those involved. This effort can easily climb to unmanageable [i.e., from the point of view of certain of the participants] proportions, especially when substantial cultural i.e., either in mathematical or in non-mathematical culture, or in both differences are involved. The current situation involving the COVID-19 pandemic adds yet another dimen- sion to the reckoning, from the point of view of the participants, of the physical and mental effort that must be expended in order to travel. As a result, when, from the point of view of at least one of the key participants, the amount of effort, time, and/or money that must be expended to travel clearly exceeds, by a substantial margin i.e., “≫” the gain [i.e., relative to various mathematical or non-mathematical criteria of the key participant in question] that appears likely to be obtained from the travel under consideration, it is highly probable that the travel under considera- tion will end up simply not taking place. One “classical” example of this phenomenon “≫” is the relative scarcity of profes- sional mathematicians in Europe or North America who travel to Japan frequently [e.g., at least once a year] or for substantial periods of time. I have, at various times in my career, been somewhat surprised by assertions on the part of some mathematicians to the effect that travel should somehow be forced on mathematicians, i.e., to the effect that some sort of coercion may somehow “override” the fundamental inequality “≫” that exists as a result of the circum- stances in which a mathematician finds him/herself in. In my experience, although this sort of coercion to travel may result in some sort of superficial influence in the very short term, it can never succeed in the long term. That is to say, the fundamental circumstances that give rise to the fundamental inequality “≫” can never be altered by means of such coercive measures to travel [cf. the discussion of [Rpt2014], (8)]. In this context, I was most impressed by the following two concrete examples, which came to my attention recently. In describing these examples, I have often used somewhat indirect expressions, in order to protect the privacy of the people involved. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 25 Example 1.9.1: The insufficiency of geographical proximity. This exam- ple concerns the results obtained in a paper written in the fall of 2019 by a graduate student (St1) from country (Ct1). This student (St1) showed his paper to a promi- nent senior researcher (Pf1) at a university in country (Ct2) in a certain area of number theory. The education and career of this researcher (Pf1) was conducted entirely at universities in countries (Ct2), (Ct3), and (Ct4). This researcher (Pf1) informed (St1) of his very positive evaluation of the originality of the results ob- tained in the paper by (St1). Another prominent senior researcher (Pf2) in a certain area of number theory was informed by (Pf1) of the paper by (St1). This researcher (Pf2), who works at a university in country (Ct2) in close physical proximity to (Pf1), also took a generally positive position with regard to the paper by (St1). On the other hand, several months subsequent to this interaction between (Pf1) and (St1), a junior researcher (Pf3), who is originally from country (Ct5), but cur- rently works at a university in country (Ct2) in close physical proximity to (Pf1) and (Pf2), informed student (St1) [via e-mail contacts between (Pf3) and (St1)’s advisor] that the results of the paper by (St1) are in fact “well-known” and essen- tially contained in papers published in the 1990’s by (Pf4), a prominent senior researcher in country (Ct6). [To be more precise, in fact the results of the paper by (St1) are not entirely contained in the papers by (Pf4) in the sense that the paper by (St1) contains certain numerically explicit estimates that are not contained in the papers by (Pf4).] Country (Ct6) is in close physical proximity to country (Ct3), and in fact, one of the research advisors of researcher (Pf4), when (Pf4) was a graduate student, was a prominent researcher (Pf5) who is originally from country (Ct7), but has pursued his career as a mathematician mainly in countries (Ct3) and (Ct6). Here, it should be pointed out that (Pf1), (Pf2), and (Pf4) are very close in age, and that (Pf1) received his undergraduate education in country (Ct3) at one of the universities that played in prominent role in the career of (Pf5). The paper by student (St1) concerns mathematics that has been studied extensively by and indeed forms one of the central themes of the research of both (Pf1) and (Pf4), but from very different points of view, using very different techniques, since (Pf1) and (Pf4) work in substantially different areas of number theory. On the other hand, at no time during the initial several months of interaction between (Pf1), (Pf2), and (St1) was the work of (Pf4) mentioned. That is to say, (Pf1) and (Pf2) discussed the results obtained in the paper by (St1) in a way that can only be explained by the hypothesis that (Pf1) and (Pf2) were, at the time, entirely unaware of the very close relationship between the results obtained in the paper by (St1) and the papers in the 1990’s by (Pf4). i.e., despite the numerous opportunities afforded by close physical prox- imity, as well as proximity of age, for substantial interaction between (Pf1) and (Pf4). The paper by student (St1) is currently submitted for publication to a cer- tain mathematical journal. Student (St1) recently received a referee’s report for his paper, which apparently [i.e., judging from the comments made in the referee’s report] was written by a mathematician working in an area of number theory close to (Pf1). This referee’s report also makes no mention of the papers in the 1990’s 26 SHINICHI MOCHIZUKI by (Pf4) and the fact that the results obtained in the paper by (St1) appear, with the exception of certain numerically explicit estimates, to be essentially contained in these papers of (Pf4). Finally, it should be mentioned that each official language of each of these countries (Ct1), (Ct2), (Ct3), (Ct4), (Ct5), (Ct6), (Ct7) belongs to the European branch of the Indo-European family of languages, and that at least six of the ten pairs of countries in the list (Ct2), (Ct3), (Ct4), (Ct6), (Ct7) share a common official language [i.e., with the other country in the pair]. Example 1.9.2: The remarkable potency of mathematical proximity. This example concerns the study of inter-universal Teichmüller theory by a graduate stu- dent (St2), who is originally from country (Ct8), but was enrolled in the doctoral program in mathematics at a university in country (Ct9) under the supervision of a senior faculty member (Pf6), who is originally from country (Ct10). This graduate student (St2) began his study of inter-universal Teichmüller theory as a graduate student and continued his study during his years as a graduate student with es- sentially no mathematical contact with any researchers who are significantly involved with inter-universal Teichmüller theory, except for his advisor (Pf6) and one mid-career researcher (Pf7) from country (Ct11). Here, we remark that the official language of each of these countries (Ct8), (Ct9), (Ct10), (Ct11) belongs to the European branch of the Indo-European family of languages. In particular, with the exception of a few very brief e-mail exchanges with me and a brief two- week long stay at RIMS in 2016 to participate in a workshop on IUT, this student (St2) had essentially no mathematical contact, prior to the fall of 2019, with any researchers at Kyoto University who are involved with inter-universal Teichmüller theory. Even in these circumstances, this student was able not only to achieve a very technically sound un- derstanding of inter-universal Teichmüller theory on his own, by reading [IUTchI-IV] and making use of various resources, activities, and contacts within country (Ct9), but also to succeed, as a graduate student, in making highly nontrivial original research contributions to a certain mild generalization of inter-universal Teichmüller theory, as well as to certain related aspects of anabelian geometry. My first [i.e., with the exception of a few very brief e-mail exchanges prior to this] mathematical contact with this student (St2) was in the fall of 2019. Although this student (St2) initially had some technical questions concerning aspects of inter- universal Teichmüller theory that he was unable to understand on his own, after a few relatively brief discussions in person with me, he was able to find answers to these technical questions in a relatively short period of time [roughly a month or two] without much trouble. §1.10. Mathematical intellectual property rights The socio-political dynamics generated by the proliferation of logically unrelated fabricated versions of inter-universal Teichmüller theory of which RCS-IUT is perhaps the most frequently cited [cf. the discussion of §1.2, as well as Examples 2.4.5, 2.4.7, 2.4.8 below] and further fueled by · grotesquely distorted mass media coverage and internet comments [cf. the discussion of §1.8], as well as by LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 27 · the conspicuous absence of detailed, explicit, mathematically substantive, and readily accessible written expositions of the logical structure underly- ing the various central assertions of the proponents of such socio-political dynamics [cf. the discussion of §1.5], have had the effect of deeply disrupting the normal process of absorption of inter-universal Teichmüller theory by the worldwide mathematical community. Left unchecked, this state of affairs threatens to pave the way for a field i.e., the field of mathematics governed by socio-political dynamics [cf. the discussion of (CmSn) in §1.5], rather than by mathematical content. From a historical point of view, various forms of institutional and concep- tual infrastructure such as the notions of · a modern judiciary system; · universal, inalienable human rights; · the rule of law; · due process of law; and · burden of proof were gradually developed, precisely with the goal of averting the outbreak of the sort of socio-political dynamics that were viewed as detrimental to society. In this context, it is interesting to note the central role played, for instance, in courts of law, by the practice of producing detailed, explicit, logically substantive, and readily accessible written doc- umentation of the logical structure underlying the various central as- sertions of the parties involved. This situation is very much reminiscent of the situation in mathematics discussed in §1.5 [cf. also the discussion of RCS-IUT in §1.3!], i.e., where we observe that it is not even possible to analyze or debate, in any sort of meaningfully definitive way, mathematical assertions such as, for instance, the historically famous assertion of Fermat to the effect that he had a proof of “Fermat’s Last Theorem”, but did not write it down in the absence of such written documentation of the logical structure of the issues under consideration. From the point of view of the above discussion, it seems natural, in the case of mathematics, to introduce, especially in the context of issues such as the one discussed above involving logically unrelated fabricated versions of inter-universal Teichmüller theory, the notion of mathematical intellectual property rights [i.e., “MIPRs”]. As the name suggests, this notion is, in some sense, modeled on the conventional notion of intellectual property rights associated, for instance, with trademarks or brand names of corporations. In the case of this conventional notion, intellectual property rights may be understood as a tool for protecting the “reli- ability” or “creditworthiness” of trademarks or brand names of a corporation from the sort of severe injury to such trademarks or brand names that may ensue from the proliferation of shoddy third-party imitations of products produced by the corporation. Here, we observe that this “severe injury” often revolves around the creation of severe obstacles to the execution of activities that play a central role in the operational normalcy of the corporation. Unlike this conventional notion, MIPRs should be understood as being as- sociated not to corporations or individuals for some finite period of time, but 28 SHINICHI MOCHIZUKI rather to mathematical notions and theories and, moreover, are of unlim- ited duration. The purpose of MIPRs may be understood as the protection of the “creditworthiness” of such a mathematical notion or theory from the severe injury to the operational normalcy of mathematical progress related to no- tion/theory that ensues from the proliferation of logically unrelated fabricated “fake” versions of the notion/theory. Before proceeding, we pause to consider one relatively elementary example of this notion of MIPRs. Example 1.10.1: The Pythagorean Theorem. (i) Recall the Pythagorean Theorem concerning the length of the hypotenuse of a right triangle in the Euclidean plane. Thus, if 0 < x y < z R are the lengths of the sides of a right triangle in the Euclidean plane, then the Pythagorean Theorem states that x 2 + y 2 = z 2 . Various versions of this result apparently may be found not only in the writings of ancient Greece and Rome, but also in Babylonian, ancient Indian, and ancient Chinese documents. Well-known “elementary proofs” of this result may be ob- tained, for instance, by computing, in various equivalent ways, the area of suitable planar regions covered by right triangles or squares that are closely related to the given right triangle. On the other hand, such “elementary proofs” typically do not address the fundamental issue of how to define such notions as length, angle, and rotation, i.e., which are necessary in order to understand the precise content of the statement of the Pythagorean Theorem. Here, we observe that if, for instance, one tries to define the notion of the length of a line segment in Euclidean space in the conventional way, then the Pythagorean Theorem reduces, in effect, to a meaning- less tautology! Moreover, although the notions of length and angle may be defined once one has defined the notion of a rotation, it is by no means clear how to give a natural definition of the notion of a rotation. For instance, one may attempt to define the notion of a rotation of Euclidean space as an element of the group generated by well-known matrices involving sines and cosines, but it is by no means clear that such a definition is “natural” or the “right definition” in some meaningful sense. Thus, in summary, it is by no means clear that such “elementary proofs” may be regarded as genuine rigorous proofs in the sense of modern mathematics. (ii) From a modern point of view, a natural, precise definition of the funda- mental notion of a rotation of Euclidean space [from which, as observed in (i), natural definitions of the notions of the notions of length and angle may be easily derived] may be given by thinking in terms of invariant tensor forms associ- ated to compact subgroups of [the topological groups determined by] various general linear groups. From this modern point of view, the precise form of the Pythagorean Theorem [i.e., “x 2 + y 2 = z 2 ”] and, in particular, the significance of the “2” in the exponent! may be traced back to the theory of Brauer groups and the closely related local class field theory of the archimedean field “R” of real numbers, i.e., in short, to various fundamental properties of the arithmetic of the topological field “R”. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 29 (iii) Considering the situation discussed in (i), (ii), it is by no means clear, in any sort of a priori or naive sense, just why the Pythagorean Theorem should take the precise form “x 2 +y 2 = z 2 ”. Indeed, since this precise form of the Pythagorean Theorem continues to appear utterly mysterious even to numerous modern-day high school students i.e., who grow up immersed in an environment replete with countless cultural links to modern mathematics, science, and technology! it seems reasonable to assume that it should have appeared all the more mysterious to the individuals who populated the various ancient civilizations mentioned in (i). In particular, it is by no means unnatural to consider the possibility that assertions similar to the following assertions [stated relative to the notation introduced in (i)] might have been made by some hypothetical individual at some time in human history: (Pyth1) “I don’t understand why the relation in the Pythagorean Theorem is of the form ‘x 2 + y 2 = z 2 ’, rather than ‘x 2 · y 2 = z 2 ’.” (Pyth2) “I would like to investigate, in the context of the Pythagorean Theorem, whether or not the relation ‘x 2 · y 2 = z 2 holds.” (Pyth3) “I investigated, in the context of the Pythagorean Theorem, whether or not the relation ‘x 2 · y 2 = z 2 holds and discovered that there exist examples that show that this relation does not in fact hold in general.” (Pyth4) “The Pythagorean Theorem is false for the following reason: The Pythagorean Theorem states that ‘x 2 · y 2 = z 2 ’, but there exist coun- terexamples that show that this relation does not hold in general.” (iv) From the point of view of the discussion given above of MIPRs, the asser- tions (Pyth1), (Pyth2), (Pyth3) do not constitute a violation of the MIPRs of the Pythagorean Theorem, but rather are precisely the sorts of assertions/comments that occur naturally in normal, sound research and educational activities in math- ematics. By contrast, (VioMIPR) (Pyth4) may be regarded as a classical example of a violation of the MIPRs of the Pythagorean Theorem. It is not difficult to imagine (DtrVio) the deeply detrimental effects on the development of mathematics throughout history that would have occurred if violations of MIPRs similar to (Pyth4) regarding the Pythagorean Theorem arose and were left unchecked. Moreover, in this context, it is important to observe that (IgNoJst) assertions of ignorance of the technical details that one must un- derstand in order to distinguish the modified version of the Pythagorean Theorem given in (Pyth4) from the original version of the Pythagorean Theorem do not by any means constitute a justification for participating in the proliferation/citation/dissemination of (Pyth4). That is to say, (BurPrf) the burden of proof of establishing any sort of logical relationship between such a modified version and the original version lies exclusively in the hands of the proponents of the modified version 30 SHINICHI MOCHIZUKI [cf. the discussion immediately following the present Example 1.10.1]. Indeed, this notion of burden of proof (BurPrf) constitutes a fundamental pillar underlying the notion of MIPRs and may be readily understood by considering the corre- sponding [perhaps more familiar] situation surrounding the conventional notion of intellectual property rights as it is typically applied to technological devices such as computers: that is to say, an assertion of technical ignorance concerning the details of the internal technical structure of a computer product manufactured by company A [or “A-product” for short] and a computer product manufactured by company X [or “X-product” for short] does not by any means [i.e., legal, ethical, or otherwise!] justify the sale, by a [say, technically ignorant] computer dealer, of an X-product advertised as an authentic A-product. (v) From a historical point of view, it appears to the author, in light of the discussion of (i), (ii), (iii), (iv), to be in some sense a sort of miracle that the Pythagorean Theorem was “discovered” in and, moreover, survived throughout the duration of numerous ancient civilizations, i.e., despite the fact that dictato- rial, authoritarian political regimes with little regard for such modern notions as a judiciary system [in the modern sense], inalienable human rights, the rule of law, due process of law, burden of proof, and so on were by no means a rarity in the ancient world. Of course, to a certain extent, this situation may be understood as a consequence of the fact that the Pythagorean Theorem is closely related to the task of direct measurement of lengths of various easily accessed [i.e., even in the ancient world!] physical objects. From this point of view of “direct measurement”, the “Pythagorean Theorem”, as understood in various ancient civilizations, should perhaps be regarded [cf. the discussion of (i), (ii)] not so much as a result in math- ematics [in the modern sense of the term], but rather as a principally empirically substantiated result in physics. Nevertheless, even when viewed from this point of view, it still seems like something of a miracle that this result survived throughout the duration of numerous ancient civilizations, unaffected by numerous meaning- less misunderstandings of the sort discussed in (iii), (iv), especially considering that similarly meaningless misunderstandings of the Pythagorean Theorem continue to plague modern-day high school students! Prior to the introduction above of the notion of MIPRs, this notion does not appear to have played an important role, at least in any sort of explicit sense, in discussions or analyses of the development of mathematics. On the other hand, Example 1.10.1 [cf., especially, Example 1.10.1, (iv)!] shows how, even at a purely implicit level, this notion of MIPRs has in fact played a fundamentally important role in the development of mathematics throughout history. Of course, in the case of violations of the MIPRs of a mathematical notion or theory, conventional courts or judiciary systems are simply not equipped to play a meaningful role in dealing with such violations, since this would require an exten- sive technical knowledge and understanding, on the part of the judges or lawyers involved, of the mathematics under consideration. Indeed, it is useful to recall in this context that · traditionally, any detrimental effects arising from such violations of the MIPRs of a mathematical notion or theory were typically averted in the LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 31 field of mathematics by means of the refereeing systems of various well established mathematical journals; · from the point of view of such traditional refereeing systems of well established mathematical journals, the burden of proof regarding the correctness of any novel assertions concerning existing mathematical no- tions or theories such as, for instance, any assertions concerning some sort of logical relationship between a modified version of a theory and the original version of the theory lies [not with the author of the original version of the theory (!), but rather] with the author of the manuscript containing the novel assertions. On the other hand, as discussed in §1.8 [cf. also the discussion of §1.3], in the case of the quite egregious MIPRs violations constituted by logically unrelated fabricated versions of inter-universal Teichmüller theory, numerous mass media re- ports and internet comments released by individuals who are clearly not operating on the basis of a solid, technically accurate understanding of the mathematics in- volved are regarded, in certain sectors of the mathematical community, as carrying much more weight than an exceptionally thorough refereeing process in a well es- tablished mathematical journal by experts on the mathematics under consideration [cf. the discussion of Example 1.5.2]. This state of affairs is deeply regrettable and should be regarded as a cause for alarm. Perhaps in the long term, new forms of institutional or conceptual infrastructure may be developed for averting the deeply detrimental effects of this sort of situation. At the time of writing, however, it appears that the only meaningful technical tool currently available to humanity for deal- ing with this sort of situation lies in the production of detailed, explicit, mathematically substantive, and readily accessible written expositions of the logical structure underlying the assertions of the various parties involved [cf. the discussion surrounding (OvDlk) in §1.5, as well as the discussion of §1.12 below], i.e., even when such assertions are purported to be a “matter of course” or “com- mon sense”, that is to say, a matter that is so profoundly self-evident that any “decent, reasonable observer” would undoubtedly find such written documentations of logical structure to be entirely unnecessary [cf. the discussion surrounding (CmSn) in §1.5]. From a historical point of view, such written documentations of logical structure can then serve as a valuable transgenerational or transcultural common core of scholarly activity a point of view that is reminiscent of the logical relator AND “∧”, which forms a central theme of the present paper. §1.11. Social mirroring of mathematical logical structure Discussions, on the part of some observers, concerning the situation surround- ing inter-universal Teichmüller theory are often dominated by various mutually exclusive and socially divisive/antagonistic dichotomies [cf. also the related discussion of §1.8], i.e., such as the following: 32 SHINICHI MOCHIZUKI (ExcDch) Is it the case that adherents of the RCS should be regarded as math- ematically correct/reliable/reasonable mathematicians, while mathemati- cians associated with inter-universal Teichmüller theory should not, OR is it the other way around? Here, the “OR” is typically understood as an “XOR”, i.e., exclusive-or. That is to say, such questions/dichotomies are typically understood as issues for which it can never be the case that an “AND” relation between the two possible alternatives under consideration holds. In discussions of mutually exclusive dichotomies such as (ExcDch), mathematicians associated with inter-universal Teichmüller theory, as well as the actual mathematical content of inter-universal Teichmüller theory, are often treated as completely and essentially disjoint entities, within the international mathematical community, from the mathematicians and mathematical research associated with the RCS. In this context, it is interesting to note that this sort of mutually exclusive dichotomy is very much reminiscent of the essential logical structure of RCS- IUT, which, as discussed in the latter portion of Example 2.4.5 below, may be understood as being essentially logically equivalent to OR-IUT, as well as to XOR- IUT, i.e., to logically unrelated fabricated versions of inter-universal Teichmüller theory in which the crucial logical AND “∧” relation satisfied by the Θ-link of inter-universal Teichmüller theory is replaced by a logical OR “∨” relation or, ˙ relation. alternatively, by a logical XOR ∨” In fact, however, (MthCnn) although it is indeed the case that the international mathematical com- munity, as well as the mathematical content of the research performed by the international mathematical community, does not consist [in the lan- guage of classical algebraic geometry] of a single irreducible component, it is nevertheless undeniably connected. In the case of inter-universal Teichmüller theory, this abundant inter-connectivity may be explicitly witnessed in the following aspects of the theory: (IntCnn1) The mathematical content of various aspects of inter-universal Teichmüller theory is closely related to various classical theories such as the following: · the invariance of heights of abelian varieties under isogeny [cf. the discussion of [Alien], §2.3, §2.4, as well as the discussion of Example 3.2.1 below; the discussion of §3.5 below]; · the classical proof in characteristic zero of the geometric ver- sion of the Szpiro inequality via the Kodaira-Spencer morphism, phrased in terms of the theory of crystals [cf. the discussion of [Alien], §3.1, (v), as well as the discussion of §3.5, §3.10 below]; · Bogomolov’s proof over the complex numbers of the geomet- ric version of the Szpiro inequality [cf. the discussion of [Alien], §3.10, (vi)]; · classical complex Teichmüller theory [cf. the discussion of Example 3.3.1 in §3.3 below]; LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 33 · the classical theory of the Jacobi identity for the theta func- tion [cf. the discussion of Example 3.3.2 in §3.3 below]; · the classical theory of the computation of the Gaussian inte- gral via polar coordinates [cf. [Alien], §3.8]. We refer to [Alien], §4, for a more detailed discussion of such relationships between inter-universal Teichmüller theory and various classical mathe- matical theories. (IntCnn2) In the context of the assertions of the RCS, it is important to recall [cf. the discussion of (MthVl) in Example 2.4.5, (viii), below] that [perhaps somewhat surprisingly!] in fact there is in some sense no disagreement among any of the parties involved with regard to the mathematical validity of the central mathematical assertions of the RCS i.e., so long as one deletes the arbitrary label “inter-universal Teichmüller theory” imposed by adherents of the RCS on the logi- cally unrelated fabricated versions [i.e., RCS-IUT/OR-IUT/XOR-IUT] of inter-universal Teichmüller theory that typically appear in discussions of adherents of the RCS [cf. the discussion of §1.2, §1.3, §1.8, §1.10]. The situation discussed in (InnCnn2) is of particular interest in the context of the present paper since the essential logical structure of this situation discussed in (InnCnn2) i.e., (CmMth) of a common mathematical understanding of the mathematical va- lidity of the various assertions under discussion, so long as one keeps track of the distinct labels “inter-universal Teichmüller theory” and “RCS- IUT/OR-IUT/XOR-IUT” is remarkably similar to the essential logical structure of the situation surrounding the central theme of the present paper, namely, the crucial logical AND “∧” property of the Θ-link in inter-universal Teichmüller theory cf. the discussion of §2.4, §3.4 below. At a more elementary level, (CmMth) may be understood as being qualitatively essentially the same phenomenon as the phenomenon discussed in Example 1.10.1, (iii), i.e., the distinction between (Pyth3) [where the distinct label “Pythagorean Theorem” is treated properly] and (Pyth4) [where the distinct label “Pythagorean Theorem” is not treated properly]. §1.12. Computer verification, mathematical dialogue, and developmental reconstruction One question that is frequently posed, in the context of the entirely unnecessary confusion that results from the plethora of misinformation and false narratives concerning inter-universal Teichmüller theory in the English-language mass media and internet [cf. the discussion of §1.8, §1.10], is the following: (CmpVer) Why not use computers to verify the mathematical validity of inter- universal Teichmüller theory? 34 SHINICHI MOCHIZUKI The implication here is that “computers” may be regarded as an entity inherently endowed with a sort of impeccable neutrality/impartiality with regard to the verifi- cation of mathematical assertions. The fundamental problem with (CmpVer) lies in the essentially tautological observation that (Algor) no computer verification algorithm for verifying some mathematical as- sertion can yield a verification of the validity of the algorithm itself, i.e., of the presumed relationship between · the mechanical output yielded by the algorithm and · the conventional human sense of “mathematical correctness” i.e., a relationship on whose integrity any sort of computer verification must be premised. Of course, in situations where the issue raised in (Algor) is not an issue of concern i.e., such as situations involving routine numerical computations or manipulation of data in some relatively simple combinatorial framework [such as a finite group, a finite simplicial complex, or a finite chain of Boolean operators] computer verifi- cation can indeed function as a meaningful tool for the verification of mathematical assertions. On the other hand, in situations such as the situation surrounding such logically unrelated fabricated versions of inter-universal Teichmüller theory as RCS-IUT in which the central issue lies [cf. the discussion of §1.2, §1.3, §1.10, as well as Example 2.4.5, 2.4.7, 2.4.8, below] in the erroneous confusion of such logically unrelated fabricated versions of a theory with the original version of the theory, computer verifications can never yield meaningful or substantive progress, since the erroneous confusion of logically unrelated fabricated versions of a theory with the original version of the theory completely invalidates, in a very essential and inevitable way, the “presumed relationship” discussed in (Algor) on which any sort of computer verification of a mathematical assertion must be premised. More elementary examples of this phenomenon of erroneous confusion of logically unrelated fabricated versions of a theory with the original version of the theory may be seen in · the situation surrounding the assertion of Example 1.10.1, (iii), (Pyth4); · the situation surrounding the [erroneous!] point of view described in Example 2.1.1 concerning the classical theory of integration on the real line i.e., situations that arise from essentially non-mathematical [e.g., social/political/ psychological] circumstances and, as a result, are clearly not amenable to resolution via computer verification. In the context of (CmpVer), it is perhaps of interest to recall from the In- troduction [cf. also the discussion at the beginning of §3.10] the point of view motivated by the well-known functional completeness, in the sense of propositional calculus, of the collection of Boolean operators consisting of logical AND “∧”, logical OR “∨”, and negation “¬” that one can, in principle, express LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 35 the essential logical structure of any mathematical argument or theory in terms of elementary logical relations, i.e., such as logical AND “∧”, logical OR “∨”, and negation “¬”. Indeed, it is precisely this point of view that formed the central motivation and conceptual starting point of the exposition given in the present paper concerning the essential logical structure of inter-universal Teichmüller theory, which may be represented symbolically as follows: A B =⇒ A (B 1 ˙ B 2 ˙ . . . ) A (B 1 ˙ B 2 ˙ . . . ˙ B 1  ˙ B 2  ˙ . . . ) =⇒ A (B 1 ˙ B 2 ˙ . . . ˙ B 1  ˙ B 2  ˙ . . . ˙ B 1  ˙ B 2  ˙ . . . ) = .. . That is to say, in summary: (SymIUT) The symbolic representation [cf. the above display!] of the essen- tial logical structure of inter-universal Teichmüller theory exposed in the present paper may be understood as being, in some sense, the closest realistic approach to the essential spirit of (CmpVer). Moreover, this symbolic representation [cf. the above display!] is sufficiently simple and transparent that, once it has been properly communicated, it may be veri- fied readily by mental computation in a matter of minutes without the use of a computer! Here, we note that the relative simplicity of this symbolic representation of (SymIUT) is obtained as a result of organizing/compartmentalizing into “blackboxes” vari- ous “blocks” of inter-universal Teichmüller theory that consist of anabelian geometry or the theory of étale theta functions, and whose validity has never arisen as a mat- ter of discussion. Ultimately, misunderstandings resulting from logically unrelated fabricated versions can only be overcome by studying the original papers [IUTchI-IV] [cf. also [Alien], as well as the present paper!] on inter-universal Teichmüller theory, or, if this is not sufficient, by engaging in constructive mathematical dialogue with mathematicians who do have a substantial, accurate understanding of the theory [cf. the discussion of §1.4, §1.5, §1.6]. Indeed, perhaps more to the point, there appears to be a conspicuous tendency, in certain sectors of the mathematical community, to (RfsDlg) utilize the proliferation/citation of logically unrelated fabricated versions of the theory as a sort of “lame excuse”/subterfuge to justify a stance of refusal to engage in such constructive mathematical dialogue concerning the theory cf. the discussion of §1.3, §1.10, especially the discussion of (VioMIPR), (DtrVio), (IgNoJst), (BurPrf) in Example 1.10.1, (iv); the discussion of Examples 3.10.1, 3.10.2 below. Moreover, (DngPrc) the situation described above in (RfsDlg), if left unchecked, constitutes, in the long-term, a dangerous precedent from the point of view of 36 SHINICHI MOCHIZUKI maintaining a state of operational normalcy in the field of mathematics in a fashion consistent with such fundamental democratic principles as the rule of law, due process of law, and burden of proof cf. the discussion of §1.10, especially the discussion of (VioMIPR), (DtrVio), (IgNoJst), (BurPrf) in Example 1.10.1, (iv). Nevertheless, in this context, it is also of fundamental importance to recall (OvrMs) the existence of numerous mathematicians [of many diverse nationali- ties!] who were indeed successful in overcoming various meaningless misunderstandings concerning inter-universal Teichmüller theory pre- cisely by persistently engaging in constructive mathematical dialogue concerning the theory. Such dialogues typically involve the painstaking and time-consuming process of (ExplLS) sifting through and analyzing the assertions of the mathematician in question concerning inter-universal Teichmüller theory in order to make precise and explicit the exact content of the logical structure under- lying such assertions [cf. the discussion of §1.5, especially Examples 1.5.1, 1.5.2]. On the other hand, it is important to emphasize that such efforts, when seen through to their conclusion, have always led to a situation in which the mathematician in question realizes his/her misunderstandings of the theory and ultimately concedes that, at least so far as he/she can see, (MthVlTh) there is indeed no mathematical reason to deny the mathematical validity of the theory. That is to say, this chain of events is precisely the chain of events that should be expected from any constructive mathematical dialogue carried out in a suitable, sincere, and rational fashion concerning a rigorously formulated mathematical the- ory. Thus, in summary, at least in a direct, literal sense, ultimately, the only way to overcome meaningless misunderstandings of the theory that arise from logically unrelated fabricated versions of the theory is to (CfrMth) directly confront the mathematical content of the original theory, either · by studying the original papers [IUTchI-IV] [cf. also [Alien], as well as the present paper!] on inter-universal Teichmüller theory, or, if this is not sufficient, · by engaging in constructive mathematical dialogue with mathe- maticians who do have a substantial, accurate understanding of the theory. [cf. the discussion of §1.3, §1.10, as well as Examples 3.10.1, 3.10.2 below]. Conversely, (BlkAcc) a stance of systematic and institutionally endorsed refusal to confront the mathematical content of the original theory has the effect of orches- trating the creation of a sort of artificial blackhole relative to the issue of LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 37 mathematical accountability [cf. the discussion of [EMSCOP] in §1.3] and leads to the sort of absurd pathologies and obstructions to the operational normalcy of the field of mathematics discussed in detail in §1.10 [cf., especially, Example 1.10.1, (iii), (Pyth4)]. That is to say, unlike the IUT community, which bears active mathematical responsibility for the mathematical content of inter-universal Teichmüller theory in the long-term by maintaining an extensive, sustained infrastructural appa- ratus of mathematical activities surrounding inter-universal Teichmüller theory such as · workshops, of one to two weeks in length, concerning inter-universal Teichmüller theory [e.g., at RIMS in March 2015, in Oxford in Decem- ber 2015, at RIMS in July 2016, and at RIMS in September 2021] and numerous lecture series [e.g., in Kumamoto in May 2014, at RIMS in De- cember 2015, in Yokohama in November 2018, and at RIMS in December 2021], which have led to a quite substantial stock of widely and readily ac- cessible PDF files of slides and videos of lectures exposing inter-universal Teichmüller theory; · extensive one-to-one mathematical discussions between mathematicians all over the world concerning inter-universal Teichmüller theory via e-mail and online video meetings; · joint research projects concerning the further development of inter- universal Teichmüller theory [cf., e.g., [ExpEst]] the author remains entirely unable, despite years of intensive effort [cf. the discussion of §1.3, §1.5, §1.6], to locate even a single mathematician who is will- ing to bear active responsibility for the mathematical content of RCS-IUT by engaging in similar mathematical activities/mathematical dialogue. The fun- damental qualitative difference constituted by this egregrious absence of an infrastructural apparatus supporting the assertions of the RCS that is to say, put another way, this sort of “hit-and-run”/“dead-end” approach to making vaguely formulated mathematical assertions that are not supported by detailed documentation/exposition apparatuses, i.e., in violation of the [EMSCOP] [cf. the discussion of the [EMSCOP] in §1.3] is precisely what is meant by the notion of a “blackhole” of mathematical accountability discussed in (BlkAcc). On the other hand, from a more long-term, historical point of view, it is perhaps of interest to observe that there is another approach to witnessing the validity of a mathematical theory, namely, (DvpRcn) the approach of developmental reconstruction, i.e., of “reconstructing the validity” of a mathematical theory by witnessing the subsequent developments that ensue from the theory. This point of view is particularly of interest in the context of inter-universal Te- ichmüller theory, given the central role played in inter-universal Teichmüller theory by anabelian geometry, i.e., which revolves around the development of recon- struction algorithms that allow one to reconstruct conventional algebraic struc- tures [i.e., of the sort that typically appear in algebraic/arithmetic geometry] from more primitive combinatorial structures such as topological groups. 38 SHINICHI MOCHIZUKI This approach of developmental reconstruction may be applied, for instance, to the task of evaluating the level of mathematical or scientific development of ancient civilizations, i.e., not via the direct study of detailed theoretical expositions [which are typically not readily available cf. the discussion of §1.5!] of the mathematics or science understood by such an ancient civilization, but rather by observing what may be understood as the “fruits” of this mathematics or science, e.g., in the form of architectural achievements such as the famous · pyramids of Egypt or · Nazca lines and mysterious ruins of Puma Punku in South Amer- ica. Another important [though non-architectural!] example of this sort of phenomenon may be seen in · the list of Pythagorean triples in the famous Babylonian tablet Plimpton 322 i.e., which is particularly notable in that it strongly suggests [that is to say, despite the fact that it is not accompanied by any sort of explicit theoretical exposition!] an understanding of algebraic manipulation on a par with the essential content of Example 1.12.1 below. Example 1.12.1: Explicit parametrization of Pythagorean triples. The set of integral solutions i.e., solutions in the ring of integers Z, also known as Pythagorean triples of the equation x 2 +y 2 = z 2 may be parametrized by applying the substitutions xz u, yz v and considering the set of rational solutions i.e., solutions in the field of rational numbers Q of the equation u 2 + v 2 = 1. The solutions of this last equation u 2 + v 2 = 1 in Q or, indeed, in any field of characteristic  = 2 may be completely parametrized by the substitutions u t t 2 −1 +1 , 2 v t 2 2t +1 where we observe that    2 t 2 −1 t 2 +1 + 2 ·   2 2t t 2 +1  = 1,  −1 = 2 · t + 1 t t =  = u+1 v  −1 u 2 +v 2 +1+2u v(u+1)  −1 = = 2 · u+1 v v + u+1  = 2 · v 1−u  = 2 · 2(u+1) v(u+1)   t 2 −1 t 2 +1    −1 2t , + 1 · t 2 +1  −1 (u+1) 2 +v 2 v(u+1)  −1 = v That is to say, this parametrization by t [for t such that t 2 + 1  = 0] gives a complete list of all solutions of the equation u 2 + v 2 = 1 [for u such that u 1  = 0] in any field of characteristic  = 2 [or, indeed, by interpreting “ = 0” as a condition of invertibility, in any ring in which 2 is invertible]. On the other hand, in this context, it is of interest to note that, at least as of the time of writing of the present paper, LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 39 (AncBas) no ancient civilization has produced evidence of knowledge of the equation  1 n=1 n 2 = π 2 6 i.e., of the solution of the so-called Basel problem. That is to say, here we note that this observation (AncBas) is valid despite the fact that each of the essential components of this equation i.e., · the positive integers of “sufficiently large value”, · the elementary operations of addition/multiplication/division, · the notion of the length of the circumference of a circle of radius 1, and · the idea of a sum of [real] numbers coming arbitrarily close, up to a very small margin of error, to another [real] number may be readily expressed in terms understandable to many advanced ancient civilizations. Of course, the discovery of an ancient civilization that produced evidence of some sort of knowledge of the equation in the display of (AncBas) would be quite startling since it would strongly suggest [that is to say, even if it is not accompanied by any sort of explicit theoretical exposition!] an understanding of numerous ideas and theorems not only from elementary differential and integral calculus, but also possibly from Fourier analysis on the circle and complex analysis on the complex plane. Finally, we return to our discussion of inter-universal Teichmüller theory. In the case of inter-universal Teichmüller theory, the phenomenon of developmental reconstruction may already be seen, albeit in a relatively weak sense, in the nu- merical results of [ExpEst]. Stronger examples of this phenomenon may be seen, however, in various enhanced versions of inter-universal Teichmüller theory that are currently under development, which are expected to give rise to various new types of applications of inter-universal Teichmüller theory. Such enhanced versions suggest strongly that the original version of inter-universal Teichmüller theory given in [IUTchI-IV] [cf. also [Alien], as well as the present paper!] should perhaps be regarded as being only the first example of a much larger collection of examples of “anabelian adèlic analysis”, i.e., in the spirit of the point of view that the various types of prime-strips that occur in inter-universal Teichmüller theory may be thought of [cf. the discussion at the end of [Alien], §3.3, (iv)] as a sort of anabelian/monoid-theoretic version of the classical notion of adèles/idèles that ap- pears throughout conventional arithmetic geometry and number theory. Here, we observe that this term “anabelian adèlic analysis” is of interest from a historical point of view in that it encapsulates, in a perhaps surprisingly efficient fashion, a quite substantial portion of the historical development of arithmetic geometry that underlies significant portions of inter-universal Teichmüller theory: indeed, (AAA0) the non-holomorphic (!) “analysis” on the real line surrounding Eu- ler’s formula [cf. the discussion of §1.5], Euler’s solution of the Basel problem [cf. the discussion above of (AncBas)], the gamma function, and the Gaussian integral [also known as the Euler-Poisson integral cf. also 40 SHINICHI MOCHIZUKI the discussion of [Alien], §3.8] i.e., analysis of the sort practiced during the 18-th century by such mathematicians as Euler and the Bernoullis may be understood as a sort of essential preparatory phase that paved the way for the holomorphic analysis/function theory of the 19-th century discussed in (AAA1), below, surrounding the theta and zeta functions [as well as the complex logarithm cf. the discussion of §1.5]; (AAA1) at a more genuine/non-preparatory level, “analysis” may be regarded as referring to the classical complex function theory pioneered by such 19-th century mathematicians as Jacobi, Riemann, and Mellin on the complex plane/upper half-plane surrounding the well-known functional equations of the theta function and the Riemann zeta function, which are related via the Mellin transform [cf. the discussion of §1.5]; (AAA2) “adèlic” may be understood as referring to the adèlization devel- oped by such 20-th century mathematicians as Chevalley, Weil, Iwa- sawa, Artin, Tate, and Langlands of the classical complex function theory of (AAA1), a development which led, in particular, to the well- known adèlic proof of the functional equation of the Dedekind zeta function [originally due to Hecke] and subsequently to the representation-theoretic approach of the Langlands program; (AAA3) “anabelian” may be interpreted as referring to the “anabelianiza- tion” of the classical functional equation of the theta function on the up- per half-plane in the fashion of inter-universal Teichmüller theory, i.e., by relating Galois groups/arithmetic fundamental groups to ring/field theory not via representation theory, as in (AAA2) (!), but rather by ap- plying cyclotomic rigidity isomorphisms and Kummer theory to relate the étale-like objects constructed by means of anabelian algorithms to their Frobenius-like counterparts arising from ring/field theory. Thus, (AAA2) may be understood as an approach to the “arithmetization” of the classical function theory of (AAA1) by means of adèlization/representation theory, i.e., by thinking, in short, of the adèles as a new domain [i.e., more pre- cisely, locally compact topological space of uncountable cardinality] in which to conduct analysis/function theory. By contrast, (AAA3) may be interpreted as an approach to the “arithmetization” of the classical function theory of (AAA1) by thinking of the abstract topological groups that arise as absolute Galois groups/arithmetic fundamental groups as the natural domain [i.e., more pre- cisely, locally compact topological space of uncountable cardinality] for conducting analysis/function theory. Thus, in summary, at the level of natural domains for conducting the analysis/function theory surrounding the theta/zeta functions, one can discern a fascinating historical progression (AAA0)  (AAA1)  (AAA2)  (AAA3) corresponding to R  C  adèles  arithmetic fundamental groups. In closing, we note that, in some sense, this interpretation of (AAA3) is consistent with the spirit of Grothendieck’s anabelian philosophy as an approach to diophantine LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 41 geometry via anabelian geometry, although, as discussed in [IUTchI], §I5, whereas Grothendieck apparently envisaged this approach as centering around the Section Conjecture, the anabelian geometry that actually appears in inter-universal Te- ichmüller theory consists mainly of absolute anabelian geometry over number fields and p-adic local fields. Section 2: Elementary mathematical aspects of “redundant copies” The essence of the central mathematical assertions of the RCS revolves, per- haps somewhat remarkably, around quite elementary considerations that lie well within the framework of undergraduate-level mathematics. Before examining, in §3, the assertions of the RCS in the technical terminology of inter-universal Teichmüller theory, we pause to give a detailed exposition of these elementary considerations. §2.1. The history of limits and integration The classical notion of integration [e.g., for continuous real-valued functions on the real line], as well as the more fundamental, but closely related notion of a limit, have a long history, dating back [at least] to the 17-th century. Initially, these notions did not have rigorous definitions, i.e., were not “well-defined”, in the sense understood by mathematicians today. The lack of such rigorous definitions frequently led, up until around the end of the 19-th century, to “contradictions” or “paradoxes” in mathematical work such as Grandi’s series  (−1) n n=0 concerning integrals or limits. Ultimately, of course, the theory of limits and integrals evolved, especially during the period starting from around the mid-19-th century and lasting until around the early 20-th century, to the extent that such “contradictions/paradoxes” could be resolved in a definitive way. This process of evolution involved, for instance, in the case of integration, first the introduction of the Riemann integral and later the introduction of the Lebesgue integral, which made it possible to integrate functions such as, for instance, the indicator function on the real line of the subset of rational numbers whose Riemann integral is not well-defined. Here, it should be noted that at various key points during this evolution of the notions of limits and integration, the central “contradictions/paradoxes” that, at times, led to substantial criticism and confusion arose from a solid, technically accurate understanding of the content and logical structure of the assertions such as, for instance, various possible approaches to computing the value of Grandi’s series at the center of these “contradictions/paradoxes”. It is precisely for this reason that such criticism and confusion ultimately lead to substantive refinements in the theory that were sufficient to resolve the original “contradictions/paradoxes” in a definitive way. Such constructive episodes in the history of mathematics which may be studied by scholars today precisely because of the existence of detailed, 42 SHINICHI MOCHIZUKI explicit, mathematically substantive, and readily accessible written records! [cf. the discussion of §1.5] stand in stark contrast to [cf. the discussion of (UndIg) in §1.3] criticism of a mathematical theory that is based on a fundamental ignorance of the content and logical structure of the the- ory, such as the following “false contradiction” in the theory of integration, which may be observed in some students who are still in an initial stage with regard to their study of the notion of integration. Example 2.1.1: False contradiction in the theory of integration. Consider the following computation of the definite integral of a real-valued function on the real line  1 x n dx = 0 1 n+1 for n a positive integer. Suppose that one takes the [drastically oversimplified and manifestly absurd, from the point of view of any observer who has an accurate understanding of the theory of integration!] point of view that the most general possible interpretation of the equation of the above display is one in which the following three symbols  1 ”, “x”, “dx” 0 are allowed to be arbitrary positive real numbers a, b, c. Here, we note that in the case of “dx”, such a substitution “dx c” could be “justified” by quoting conven- tional “-δ” treatments of the theory of limits and integrals, in which infinitesimals i.e., such as “dx” or “” are allowed to be arbitrary positive real numbers, which are regarded as being “arbitrarily small”, and observing that any positive real number c is indeed much smaller than “most other positive real numbers” [such as 1000 · c, etc.]. On the other hand, by substituting the values n = 1, 2, 3, one obtains relations abc = 1, ab 2 c = 12 , ab 3 c = 13 . The first two of these relations imply that b = 12 [so b 2 = 14 ], while the first and third relations imply that b 2 = 13  = 14 a “contradiction”! §2.2. Derivatives and integrals In the context of the historical discussion of integration in §2.1, it is interesting to recall the fundamental theorem of calculus, i.e., the result to the effect that, roughly speaking, the operations of integration and differentiation of functions [i.e., real-valued functions on the real line satisfying suitable conditions] are inverse to one another. Thus, from a certain point of view, the “essential information” contained in a function may be understood as being “essentially equivalent” to the “essential information” contained in the derivative of the function that is to say, since one may always go back and forth at will between a function and its derivative by integrating and then differentiating. This point of view might then tempt some observers to conclude that LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 43 any mathematical proof that relies, in an essential way, on consideration of the derivative of a function must be fundamentally flawed since any information that might possibly be extracted from the derivative of the function should already be available [cf. the “essential equivalence” dis- cussed above] from the function prior to passing to the derivative, i.e., in “contradiction” to the essential dependence of the proof on passing to the derivative. Alternatively, this point of view may be summarized in the following way: the “essential equivalence” discussed above implies that any usage, in a mathematical proof, of the derivative of a function is necessarily inherently redundant in nature. In fact, of course, such “pseudo-mathematical reasoning” is itself fundamen- tally flawed. Two examples of well-known proofs in arithmetic geometry that depend, in a essential way, on passing to the derivative will be discussed in the final portion of §3.2 below. These examples are in fact closely related to the mathematics that inspired inter-universal Teichmüller theory [cf. the discussion in the final por- tion of §3.2 below]. One central aspect of the situations discussed in §3.2 below is the exploitation of properties of [various more abstract analogues of] the derivative of a function that differ, in a very substantive, qualitative way, from the prop- erties of the original function. One important example of this sort of situation is the validity/invalidity of various symmetry properties. This phenomenon may be observed in the following elementary example. Example 2.2.1: Symmetry properties of derivatives. The real-valued func- tion f (x) = x on the real line is not invariant [i.e., not symmetric] with respect to translations by an arbitrary constant c R. That is to say, in general, it is not the case that f (x + c) = f (x). On the other hand, the derivative f  (x) = 1 of this function is manifestly invariant/symmetric with respect to such trans- lations. §2.3. Line segments vs. loops By comparison to the examples given in §2.1, §2.2, the following elementary geometric examples are much more closely technically related to the assertions of the RCS concerning inter-universal Teichmüller theory. Example 2.3.1: Endpoints of an oriented line segment. (i) Write def I = [0, 1] R 44 SHINICHI MOCHIZUKI for the closed unit interval [i.e., the set of nonnegative real numbers 1] in the real line R. Thus, I is equipped with a natural topology [i.e., induced by the topology of R], hence can be regarded as a topological space, indeed more specifically, as a one-dimensional topological manifold with boundary that is equipped with a natural orientation [i.e., induced by the usual orientation of R]. Write def α = {0}, def β = {1} for the topological spaces [consisting of a single point!] determined by the two endpoints of I. Thus, α and β are isomorphic as topological spaces. In certain situations that occur in category theory, it is often customary to replace a given category by a full subcategory called a skeleton, which is equivalent to the given category, but also satisfies the property that any two isomorphic objects in the skeleton are equal. This point of view of working with skeletal categories [i.e., categories which are their own skeletons] is motivated by the idea that nonequal isomorphic objects are “redundant”. Of course, there are indeed various situations in which nonequal isomorphic objects are redundant in the sense that working with skeletal categories, as opposed to arbitrary categories, does not result in any substantive difference in the mathematics under consideration. (ii) On the other hand, if, in the present discussion of I, α, β which one may visualize as follows I •——>——• α β one identifies α and β, then one obtains a new topological space, that is to say, more specifically, an oriented one-dimensional topological manifold [whose orienta- tion is induced by the orientation of I] def L = I/ α β that is homeomorphic to the unit circle, i.e., may be visualized as a loop. Write γ L L for the image of α I, or, equivalently, β I, in L. As is well-known from elementary topology, the topological space L is structurally/qualitatively very different from the topological space I. For instance, whereas I has a trivial fun- damental group, L has a nontrivial fundamental group [isomorphic to the additive group of integers Z]. In particular, it is by no means the case that the fact that α and β are isomorphic as topological spaces implies a sort of “redundancy” to the effect that any mathematical argument involving I [cf. the above observation concerning fundamental groups!] is entirely equivalent to a corresponding math- ematical argument in which α and β are identified, i.e., in which “I” is replaced by “L”. (iii) In this context, we observe that the [one-dimensional oriented topological manifold with boundary] I does not admit any symmetries that switch α and LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 45 β. Moreover, even if one passes to the quotient I  L, the [one-dimensional ori- ented topological manifold] L does not admit any symmetries that reverse the orientation of L. Example 2.3.2: Gluing of adjacent oriented line segments. (i) A similar elementary geometric situation to the situation discussed in Ex- ample 2.3.1, but which is technically a bit more similar to the situation that arises in inter-universal Teichmüller theory may be given as follows. We begin with two distinct copies I, I of I. Thus, I has endpoints α, β [i.e., corresponding re- spectively to the endpoints α, β of I]; similarly, I has endpoints α, β. We then proceed to form a new topological space J by gluing I to I via the unique isomor- phism of topological spaces β α. Thus, β and α are identified in J. Let us write γ J for the one-pointed topological space obtained by identifying β and α. Thus, J may be visualized as follows: I I ——>——•——>—— α γ J β (ii) Observe that the gluing operation that gave rise to J is such that we may regard I and I as subspaces I J, I J of J. Since each of these subspaces I, I of J is naturally isomorphic to I, one may take the point of view, as in the discussion of Example 2.3.1, that these two subspaces are “redundant” and hence should be identified with one another [say, via the natural isomorphisms of I, I with I] to form a new topological space M = J/ I I def where we observe that the natural isomorphisms of I, I with I determine a natural isomorphism of topological spaces M L = I/ α β, with the loop L considered in Example 2.3.1. Write γ M M for the image of γ J J in M. Thus, the natural isomorphism M L maps γ M isomorphically onto γ L . On the other hand, just as in the situation discussed in Example 2.3.1, it is by no means the case that the fact that I and I are [in fact, natu- rally] isomorphic as topological spaces implies a sort of “redundancy” to the effect that any mathematical argument involving J is entirely equiv- alent to a corresponding mathematical argument in which I and I are identified [say, via the natural isomorphisms of I, I with I], i.e., in which “J” is replaced by “M”. Indeed, for instance, one verifies immediately, just as in the situation of Exam- ple 2.3.1, that the fundamental groups of J and M are not isomorphic. That is to say, just as in the situation of Example 2.3.1, the topological space J is struc- turally/qualitatively very different from the topological space M. 46 SHINICHI MOCHIZUKI §2.4. Logical AND “∧” vs. logical OR “∨” The essential mathematical content of the elementary geometric examples dis- cussed in §2.3 may be reformulated in terms of the symbolic logical relators AND “∧” and OR “∨”. This reformulation renders the elementary geometric examples of §2.3 in a form that is even more directly technically related to various central aspects of the assertions of the RCS concerning inter-universal Teichmüller theory. Example 2.4.1: “∧” vs. “∨” for adjacent oriented line segments. (i) Recall the situation discussed in Example 2.3.2. Thus, J I β = γ J = α I J, i.e., (AOL1) the following condition holds:   γ J = β I   γ J = α I . On the other hand, if one identifies I, I, then one obtains a topological space M L, i.e., a loop. Here, denotes the natural isomorphism discussed in Example 2.3.2, (ii). Now suppose that we are given a connected subspace γ I I whose image in the quotient I  L = I/ α β coincides with γ L L, i.e., with the image of γ J J via the composite of the quotient J  M = J/ I I with the natural isomorphism M L. Then observe that (AOL2) the following condition holds: γ I {α, β}, i.e.,   γ I = β I   γ I = α I . Of course, (AOL3) the essential mathematical content discussed in this condition (AOL2) may be formally described as a condition involving the AND relator “∧”:   β {α, β}   α {α, β} . But the essential mathematical content of the OR relator “∨” statement in (AOL2) remains unchanged. (ii) On the other hand, [unlike the case with γ J !] (AOL4) the following condition does not hold:   γ I = β I   γ I = α I . LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 47 That is to say, in summary, the operation of identifying I, I e.g., on the grounds of “redundancy” [cf. the discussion of Example 2.3.2] has the effect of passing from a situation in which the AND relator “∧” holds [cf. (AOL1)] to a situation in which the OR relator “∨” holds [cf. (AOL2), (AOL3)], but the AND relator “∧” does not hold [cf. (AOL4)]! (iii) It turns out that this phenomenon i.e., of an identification of “redundant copies” leading to a passage from the validity of an “∧” relation to the validity of an “∨” relation coupled with the invalidity of an “∧” relation forms a very precise model of the situation that arises in the assertions of the RCS concerning inter-universal Teichmüller theory [cf. the discussion of §3.2, §3.4 below]. Example 2.4.2: Differentials on oriented line segments. (i) In the situation of Example 2.4.1, one way to understand the gap between (AOL1) and (AOL4) i.e., the central issue of whether the AND relator “∧” holds or does not hold is to think in terms of the restriction to I R of the coordinate function “x” of Example 2.2.1. Indeed, (AOD1) one may interpret (AOL4) as the statement that the coordinate functions “x” on the two copies I, I that constitute J do not glue together to form a single, well-defined R-valued function on J [that is to say, since it is not clear whether the value of such a function on γ J J should be 0 or 1, i.e., such a function is not well-defined on γ J J]; (AOD2) on the other hand, (AOL1) may be interpreted as the statement that such a function [i.e., obtained by gluing together the coordinate functions “x” on the two copies I, I that constitute J] can indeed be defined if one regards its values as being [not in a single copy of R, but rather] in the set †,‡ R obtained by gluing together two distinct copies R, R of R by identifying 1 R with 0 R. (ii) On the other hand, if, instead of considering the coordinate function “x”, one considers the differential “dx” associated to this coordinate function [cf. the discussion of Example 2.2.1], then one observes immediately that (AOD3) the differentials “dx” on the two copies I, I that constitute J do indeed glue together to form a single, well-defined differential on J that, moreover, is compatible with the quotient J  M = J/ I I in the sense that, as is easily verified, it arises as the pull-back, via this quotient map J  M, of a [smooth] differential on the [smooth manifold constituted by the] loop M. Note, moreover, that the gluings and compatibility of (AOD3) may be achieved without considering functions or differentials valued in some sort of complicated [i.e., by comparison to R!] set such as †,‡ R. 48 SHINICHI MOCHIZUKI (iii) It turns out [cf. the discussion of Example 2.4.1, (iii)] that the phenomenon discussed in (AOD3) is closely related to the situation that arises in inter-universal Teichmüller theory [cf. the discussion of §3.2 below]. Example 2.4.3: Representation via subgroup indices of “∧” vs. “∨”. (i) Let A be an abelian group and B 1 , B 2 A subgroups of A such that B 1 B 2 has finite index in B 1 and B 2 . Then one may define a positive rational number, which we call the index of B 2 relative to B 1 , def [B 1 : B 2 ] = [B 1 : B 1 B 2 ]/[B 2 : B 1 B 2 ] Q >0 . Thus, [B 1 : B 2 ] · [B 2 : B 1 ] = 1; when B 2 B 1 , this notion of index coincides with the usual notion of the index of B 2 in B 1 . (ii) Let n be a positive integer 2. Consider the diagram of group homomor- phisms G 1 −→ G 2 −→ G 3 where, for i = 1, 2, 3, G i denotes a copy of [the additive group of rational integers] Z, and the arrows are given by multiplication by n. For i = 1, 2, 3, write G Q i = G i Z Q for the tensor product of G i over Z with Q. Then observe that this diagram induces a diagram of group isomorphisms def G Q 1 G Q 2 G Q 3 i.e., in which the arrows are isomorphisms. Let us use these isomorphisms to Q identify the groups G Q i , for i = 1, 2, 3, and denote the resulting group by G . (iii) Observe that the first diagram of (ii) is structurally reminiscent of the object J discussed in Examples 2.3.2, 2.4.1, 2.4.2, i.e., if one regards · the first arrow of the first diagram of (ii) as corresponding to I, · the second arrow of the first diagram of (ii) as corresponding to I, and · G 1 , G 2 , and G 3 as corresponding to α, β = α, and β, respectively. Here, we observe that G 2 appears simultaneously as the codomain of the arrow G 1 −→ G 2 AND [cf. (AOL1)!] as the domain of the arrow G 2 −→ G 3 . Moreover, we may consider indices of G 1 , G 2 , and G 3 as subgroups of G Q [G 2 : G 1 ] = [G 1 : G 2 ] −1 = n; [G 3 : G 2 ] = [G 2 : G 3 ] −1 = n; [G 3 : G 1 ] = [G 1 : G 3 ] −1 = n 2 in a consistent fashion, i.e., in a fashion that does not give rise to any contra- dictions. (iv) On the other hand, suppose that we delete the “distinct labels” G 1 , G 2 , G 3 from the copies of Z considered in the first diagram of (ii). This yields a diagram Z −→ Z −→ Z LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 49 in which the second arrow may be regarded as a copy of the first arrow. This situation might motivate some observers to conclude that these two arrows are “redundant” and hence should be identified with one another cf. the discus- sion of the quotient J  M in Example 2.3.2, (ii) to form a diagram Z consisting of a single copy of Z and the endomorphism of this single copy of Z given by multiplication by n. At first glance, this operation of identification may appear to give rise to various “contradictions” in the computation of the index, i.e., such as 1 = [G 1 : G 1 ] = [Z : Z] = [G 2 : G 1 ] = n 2 and so on. In fact, however, if one takes into account the OR relator “∨” [but not the AND relator “∧”!] relations that one obtains upon executing the identification operation in question [cf. (AOL2), (AOL4)!], then one concludes that [after exe- cuting the identification operation in question!] each of the indices [G i : G j ], for i, j {1, 2, 3}, may only be computed up to multiplication by an integral power of n, i.e., that each index [G i : G j ], for i, j {1, 2, 3}, is only well-defined as “some def indeterminate element” of n Z = {n m | m Z} Q >0 . In particular, in fact there is no contradiction. Example 2.4.4: Logical “∧/∨” vs. “narrative ∧/∨”. Consider the following argument concerning a natural number x {1, 3}, i.e., a natural number for which it holds that (x = 1) (x = 3): (Nar1) Suppose that x = 3. Then it follows that x = 3 > 2. That is to say, we conclude that x > 2. (Nar2) Since (x = 1) (x = 3), we may consider the case x = 1. Then, by applying the conclusion of (Nar1), we conclude that 1 = x > 2, i.e., that 1 > 2 a contradiction! Of course, this argument is completely fallacious! On the other hand, it yields a readily understood concrete example of the absurdity that arises when, as is in effect done in (Nar2), logical OR “∨” is confused with logical AND “∧”! In various contexts, this sort of confusion can arise from the ambiguity of various narrative expressions that appear in the discussion of a mathematical argument. This sort of ambiguity can lead to a situation in which a “narrative AND ∧” i.e., the fact that in a particular narrative exposition of an argument, one performs both the task of considering the case “x = 3” [cf. (Nar1)] and the task of considering the case “x = 1” [cf. (Nar2)] is mistakenly construed as a logical AND “∧”. In a similar vein, one may consider situations in which the roles played by “∧” and “∨” are reversed, i.e., in which a “narrative OR ∨” i.e., the fact that in a particular narrative exposition of an argument, one’s attention is concentrated 50 SHINICHI MOCHIZUKI either on the task of considering one situation or on the task of considering another situation is mistakenly construed as a logical OR “∨”. Indeed, it appears that one fundamental cause, in the context of the essential logical structure of inter-universal Teichmüller theory [cf. the discussion of Example 2.4.5 below!], of the confusion on the part of some mathematicians between logical AND “∧” and logical OR “∨” lies precisely in this sort of confusion between “narrative ∧/∨” and logical “∧/∨”. Example 2.4.5: Numerical representation of “∧” vs. “∨”. (i) A slightly more sophisticated numerical representation of the difference between “∧” and “∨” which in fact mirrors the essential logical structure of inter-universal Teichmüller theory in a very direct fashion may be given as follows [cf. [Alien], Example 3.11.4]. Indeed, the essential logical flow of inter- universal Teichmüller theory may be summarized as follows: · one starts with the definition of an object called the Θ-link; · one then constructs a complicated apparatus that is referred to as the multiradial representation of the Θ-pilot [cf. [IUTchIII], Theorem 3.11]; · finally, one derives a final numerical estimate [cf. [IUTchIII], Corol- lary 3.12] in an essentially straightforward fashion from the multiradial representation of the Θ-pilot. (ii) An elementary model of this essential logical flow may be given by means of real numbers A, B R >0 and , N R such that 0  1 in the following way: · Θ-link:   def N = −2B   N = −A ; def · multiradial representation of the Θ-pilot:   N = −2A +    N = −A ; · final numerical estimate: −2A +  = −A, hence A = , i.e., A 1. Thus, the definition of the Θ-link and the construction of the multiradial represen- tation of the Θ-pilot are meaningful/nontrivial precisely on account of the validity of the AND relator “∧”, which is rendered possible, in the definition of the Θ-link, precisely by allowing the real numbers A, B to be [a priori] distinct real numbers cf. (AOL1) vs. (AOL4), where we think in terms of the correspondences B −2B ←→ ←→ I, N ←→ γ J , A β, −A ←→ −2A α, ←→ ←→ I β. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 51 The passage from the multiradial representation of the Θ-pilot to the final numerical estimate is then immediate/straightforward/logically transparent. (iii) By contrast, if, in the elementary numerical model of (ii), one replaces “∧” by “∨”, then our elementary numerical model of the logical structure of inter- universal Teichmüller theory takes the following form: · “∨” version of Θ-link:     def def N = −A N = −2B   def [cf. N = −2A   N = −A ]; def · “∨” version of multiradial representation of the Θ-pilot:     N = −2A +  N = −A ; · final numerical estimate: −2A +  = −A, hence A = , i.e., A 1. That is to say, the use of distinct real numbers A, B in the definition of the “∨” ver- sion of Θ-link seems entirely superfluous [cf. (AOL2), relative to the correspon- dences discussed in (ii)]. This motivates one to identify A and B i.e., to suppose “for the sake of simplicity” that A = B which then has the effect of rendering the definition of the original “∧” version of the Θ-link invalid/self-contradictory [cf. (AOL4), relative to the correspondences discussed in (ii)]. Once one identifies A and B, i.e., once one supposes “for the sake of simplicity” that A = B, the pas- sage from the “∨” version of Θ-link to the resulting “∨” version of the multiradial representation of the Θ-pilot then seems entirely meaningless/devoid of any interesting content. The passage from the resulting meaningless “∨” version of the multiradial representation of the Θ-pilot to the final numerical estimate then seems abrupt/mysterious/entirely unjustified, i.e., put another way, looks as if one erroneously replaced the “∨” in the meaningless “∨” version of the multiradial representation of the Θ-pilot by an “∧” without any mathematical justification whatsoever. It is precisely this pernicious chain of misunderstandings emanating from the “redundancy” assertions of the RCS that has given rise to a substantial amount of unnecessary confusion concerning inter-universal Teichmüller theory. (iv) Before proceeding, we observe that the sort of confusion discussed in (iii) between “∧” and “∨” can occur as the result of any of the following phenomena: (AOC1) a confusion between “narrative ∧/∨” and logical “∧/∨”, as discussed in Example 2.4.4; (AOC2) thinking in terms of the “fake ∧” of (AOL3), i.e., which, though for- mulated as a logical AND “∧” relation, is in fact, substantively speaking, a logical OR “∨” relation; 52 SHINICHI MOCHIZUKI (AOC3) the symptom (Syp2) discussed in §3.6 below, i.e., a desire to see the “proof” of some sort of commutative diagram or “compatibility prop- erty” to the effect that taking log-volumes of pilot objects in the domain and codomain of the Θ-link yields the same real number; (AOC4) a fundamental misunderstanding which is often closely inter- twined with the symptom (Syp2) discussed in (AOC3) of the meaning of the crucial closed loop of §3.10, (Stp7), (Stp8), below [cf. §3.10, (Stp7), (Stp8), as well as the following discussion]. (v) Let us refer to the “∧” version of inter-universal Teichmüller theory dis- cussed in (ii) i.e., the original version of inter-universal Teichmüller theory, in which one interprets the Θ-link as a logical AND “∧” relation as AND-IUT. Thus, AND-IUT = IUT is the original version of inter-universal Teichmüller theory. Let us refer to the “∨” version of inter-universal Teichmüller theory discussed in (iii) i.e., the version of inter-universal Teichmüller theory that arises if one [mis- takenly!] interprets the Θ-link as a logical OR “∨” relation as OR-IUT. As discussed in (iii), in OR-IUT, one is motivated to implement the RCS-identifications of RCS-redundant copies of objects i.e., in the language of (iii), to “identify A and B” and hence to conclude that OR-IUT =⇒ RCS-IUT, where we recall that “RCS-IUT” refers to the version of inter-universal Teichmüller theory obtained by implementing the RCS-identifications of RCS-redundant copies of objects [cf. the discussion of §1.2]. On the other hand, it is not difficult to see that in RCS-IUT, one is forced to work with a (NeuORInd) indeterminacy [cf. the discussion at the end of §3.4 below, as well as the discussion of (ΘORInd) in §3.11 below], i.e., to interpret ˙ relation [that is to say, a logical OR “∨” the Θ-link as a logical XOR ∨” relation such that the corresponding logical AND “∧” relation cannot hold cf. the discussion of (iii)]. In particular, we conclude that RCS-IUT =⇒ XOR-IUT =⇒ OR-IUT [where the second =⇒ is a consequence of well-known general properties of Boolean operators], i.e., in summary: (XOR/RCS) we have equivalences XOR-IUT ⇐⇒ OR-IUT ⇐⇒ RCS-IUT. In the following, I shall refer to the school of thought [i.e., in the sense of a “collection of closely interrelated ideas”] surrounding OR-IUT as ORS, i.e., the “OR school [of thought]”, and to the school of thought surrounding XOR-IUT as XORS, i.e., the “XOR school [of thought]”. Thus, XORS = ORS = RCS. (vi) On the other hand, one may also consider yet another version of inter- universal Teichmüller theory, also motivated by the discussion of (iii), which we refer to as EssOR-IUT, i.e., “essentially OR IUT”. This is the version of inter- universal Teichmüller theory in which one accepts, at the level of formal definitions, the logical AND “∧” version of the Θ-link as in (ii), i.e., without identifying A and B, but [for some unexplained reason!] one then arbitrarily shifts, when considering the multiradial representation of the Θ-pilot, to the logical OR “∨” interpretation of the multiradial representation of the Θ-pilot, i.e., as in (iii). That is to say, as the name “EssOR-IUT” suggests, the fundamental logical AND LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 53 “∧” property of the Θ-link is never actually used in any sort of meaningful way in EssOR-IUT. In particular, (EssOR/RCS) although, at a purely formal level, EssOR-IUT rejects RCS-IUT, the essential logical structure of EssOR-IUT still nevertheless gives rise to the abrupt/mysterious/entirely unjustified transition discussed in (iii) to the final numerical estimate. It appears that the “arbitrarily shift” referred to above is often precipitated by the various phenomena discussed in (iv) [cf., especially, (AOC3), (AOC4)]. In the following, I shall refer to the school of thought [i.e., in the sense of a “collection of closely interrelated ideas”] surrounding EssOR-IUT as EssORS, i.e., the “essen- tially OR school [of thought]”. (vii) In general, at the level of formalities of Boolean operators, “∧ ∨”, but “∨ ⇒ ∧”. In particular, in the context of the transition to the final numerical estimate of inter-universal Teichmüller theory, (∨ ⇒ ∧) it appears entirely hopeless/unrealistic to pass from the “∨” version of the multiradial representation of the Θ-pilot to the “∧” version of the multiradial representation of the Θ-pilot. This is precisely the “abrupt/mysterious/entirely unjustified” transition [to the final numerical estimate] discussed in (iii). (viii) The discussion of (v), (vi), (vii) may be summarized as follows [cf. also the discussion of §1.2, §1.3]: · The fundamental misunderstanding on the part of adherents of the RCS = ORS = XORS to the effect that OR-IUT or XOR-IUT is indeed the content of AND-IUT = IUT leads to the mistaken interpretation of the assertion (XOR/RCS) as an equivalence between AND-IUT = IUT and RCS-IUT. · The fundamental misunderstanding on the part of adherents of the RCS = ORS = XORS to the effect that OR-IUT or XOR-IUT is indeed the content of AND-IUT = IUT leads to the mistaken interpretation of the assertion (∨ ⇒ ∧) as a logical flaw in AND-IUT = IUT. · The fundamental misunderstanding on the part of adherents of the EssORS to the effect that EssOR-IUT is indeed the content of AND-IUT = IUT leads either to the mistaken interpretation of the assertion (∨ ⇒ ∧) as a logical flaw in AND-IUT = IUT or to the mistaken interpretation of the assertion (∨ ⇒ ∧) as an indication the existence of some sort of infinitely complicated and mysterious argument i.e., for concluding that “∨ ∧”! in inter-universal Teichmüller theory that requires years of concerted effort to understand. Thus, as the descriptive “essential” suggests, there is in fact, from the point of view of the essential logical structure under consideration, very little difference between EssORS and XORS = ORS = RCS or between EssOR-IUT and XOR-IUT ⇐⇒ OR-IUT ⇐⇒ RCS-IUT. · In fact, the correct interpretation of the assertion (∨ ⇒ ∧) consists of the conclusion that neither XOR-IUT ⇐⇒ OR-IUT ⇐⇒ RCS-IUT nor 54 SHINICHI MOCHIZUKI EssOR-IUT has any direct logical relationship to AND-IUT = IUT. Assertions of various schools of thought Actual mathematical content RCS = ORS = XORS: “IUT RCS-IUT” “XOR-IUT OR-IUT RCS-IUT” RCS = ORS = XORS: “IUT is logically flawed.” EssORS: either “IUT is logically flawed.” or “The logical structure of IUT is infinitely complicated/mysterious.” “∨ ⇒ ∧”, which implies that “(AND-IUT =) IUT ⇒ RCS-IUT”, “(AND-IUT =) IUT ⇒ OR-IUT”, “(AND-IUT =) IUT ⇒ XOR-IUT” “∨ ⇒ ∧”, which implies that “(AND-IUT =) IUT ⇒ EssOR-IUT” Here, we observe that the above analysis is in some sense remarkable in that it makes explicit the fact that, if one forgets the arbitrary label “inter-universal Te- ichmüller theory” placed on XOR-IUT or OR-IUT or EssOR-IUT by adherents of the RCS = ORS = XORS or the EssORS, then [perhaps somewhat surprisingly!] (MthVl) there is in fact no disagreement among any of the parties involved with regard to the mathematical validity of the mathematical assertions (XOR/RCS) and (∨ ⇒ ∧). Indeed, this state of affairs may be understood as in some sense highlighting the es- sentially social/political/psychological, i.e., in summary, non-mathematical nature of the entirely unnecessary confusion that has arisen concerning inter-universal Te- ichmüller theory [cf. the discussion of §1.8, §1.11]. The above observations are summarized in the “dictionary of assertions” given above. Example 2.4.6: Carry operations in arithmetic, geometry, and Boolean logic. (i) Observe that if, in the situation of Example 2.4.3, (ii), one focuses one’s attention on the subset D 4−i G i in the copy of Z denoted by G i , where i = 1, 2, 3, corresponding to {0, 1, 2, . . . , n 1} Z, then the situation considered in Example LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 55 2.4.3, (ii), closely resembles the situation that arises in elementary arithmetic com- putations such as addition and multiplication involving base n expansions of natural numbers. That is to say, one may think of · D 1 as the first digit, i.e., when n = 10, the “ones digit”, · D 2 as the second digit, i.e., when n = 10, the “tens digit”, and · D 3 as the third digit, i.e., when n = 10, the “hundreds digit” of such an expansion. When performing such elementary arithmetic computations such as addition and multiplication involving base n expansions of natural numbers, recall that it is of fundamental importance to take into account the various carry operations that occur. In particular, we observe that the use of distinct labels for distinct digits plays a fundamental role in elementary arithmetic computations involving base n expansions of natural numbers cf. the distinct labels “G 1 ”, “G 2 ”, “G 3 in the discussion of Example 2.4.3, (ii), (iii). This situation is reminiscent of the important role played by the distinct labels “A”, “B” in the “Θ-link” of Example 2.4.5, (ii). Note, morever, that deletion/confusion of these distinct labels for distinct digits has the effect of completely invalidating, at least in the usual “strict sense”, elementary arithmetic computations involving base n expansions of natural numbers cf. the situation considered in Example 2.4.3, (iv). On the other hand, if one restricts one’s attention to a specific computational algorithm [involving, say, addition and multiplication operations], then in fact it is often possible i.e., de- pending on the content of the specific computational algorithm under consideration to obtain estimates to the effect that applying the algorithm either with or without the use of distinct labels for distinct digits in the base n expansions of the natural numbers involved in fact yields the same result, up to some explicitly bounded discrepancy. [For instance, when n = 10, any algorithm that only involves addition and multipli- cation operations yields the same result modulo 9 (= 10 1), regardless of whether or not one uses distinct labels for distinct digits in decimal expansions of natural numbers.] Such estimates are reminiscent of the “multiradial representation” of Example 2.4.5, (ii). (ii) Observe that the discussion of · adjacent oriented line segments I”, I” and · oriented loops “L”, “M” in Examples 2.3.1, 2.3.2, 2.4.1, 2.4.2 [cf. also the discussion of Examples 2.4.3, (iii); 2.4.5, (ii)] may be regarded as a sort of limiting case of the discussion of base n expansions of natural numbers in (i) above, i.e., if one · considers the real numbers obtained by dividing the natural numbers 2n in the discussion of (i) above by n and then 56 SHINICHI MOCHIZUKI · passes to the limit n +∞. That is to say, in summary, the adjacency of the oriented line segments I”, I” may be under- stood as a sort of continuous, geometric representation of the carry operation that appears in elementary arithmetic computations involving base n expansions of natural numbers. (iii) From the point of view of discussions of the logical structure of math- ematical arguments represented in terms of Boolean operators such as logical AND “∧” and logical OR “∨”, it is of interest to consider the discussion of (i) above in the binary case, i.e., the case n = 2. We begin our discussion of the binary case by recalling the following well-known facts: · multiplication in the field F 2 = {0, 1} may be regarded as corresponding to the Boolean operator AND “∧”; · addition in the field F 2 = {0, 1} may be regarded as corresponding to the Boolean operator XOR [i.e., “exclusive OR”], which we denote by ˙ ∨”; · “carry-addition” in the truncated ring of Witt vectors F 2 × F 2 i.e., addition of two elements of F 2 {0} × F 2 F 2 × F 2 , regarded as Teichmüller representatives in the truncated ring of Witt vectors Z/4Z, that is to say, an addition operation in which one allows for the carry operation [cf. the discussion of (i)!] to the first factor of F 2 × F 2 may be regarded as corresponding to an operator that we shall refer to as the ¨ ”; thus, we have “COR”, or “carry-OR”, operator and denote by ¨ = (∧, ∨) ˙ ¨ 0 = (0, 0); 1 ¨ 0 = 0 ¨ 1 = (0, 1); 1 ¨ 1 = (1, 0)]. [so 0 These well-known facts case may be summarized as follows: ¨ = ∨) ˙ Conventional mixed-characteristic/“carry” addition in Z consid- ( ¨ may be described in terms of the “splitting” ered modulo 4 i.e., of the natural surjection Z  F 2 determined by the “Teichmüller repre- sentatives” 0, 1 Z via the equation ¨ = (∧, ∨) ˙ ¨ as an operation obtained by “stacking” mul- i.e., which exhibits ˙ in F 2 . Here, we note tiplication “∧” in F 2 on top of addition ∨” that this splitting via Teichmüller representatives 0, 1 Z is compatible with the multiplicative structures in Z and F 2 , but not with the addi- tive structures in Z and F 2 . Put another way, one may think of the ring structures of Z and F 2 as structures that share a common multiplicative structure [cf. “∧”!], but do not share a common additive structure [cf. ˙ ∨”!]. These observations will be of fundamental importance in the theory developed in §3 [cf., especially, the discussion at the beginning of §3.10]. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 57 (iv) Some readers may object to the comparisons and analogies between inter- universal Teichmüller theory and the “mathematics of carry operations” dis- cussed in (i), (ii), and (iii) as being inappropriate on the grounds that this mathe- matics of carry operations is much too “trivial” to be of any substantive interest. On the other hand, we observe that the mathematics of carry operations discussed in (i), (ii), and (iii) is intimately intertwined with numerous important develop- ments in the history of mathematics. With regard to the content of (i), we recall that the use of place-value decimal numerals, i.e., that make use of notation for zero, appears to date back to Indian texts and inscriptions from the 7-th to 9-th centuries AD. Such numerals also reached the Arab world during this period, but this Hindu-Arabic numeral system apparently only became widely used in Europe during the late middle ages, between the 13-th and 15-th centuries. In this context, one noteworthy development was the book Liber Abaci published by the Italian mathematician Fibonacci in 1202, which promoted the use of the Hindu- Arabic numeral system in Europe. Here, it is useful to recall that, by comparison to earlier numeral systems, such as the Greco-Roman and Babylonian systems, place- value decimal numerals not only facilitate elementary arithmetic computations i.e., via the systematic use of carry operations, as discussed in (i)! but also make it possible to express all hence, in particular, infinitely many natural numbers by means of finitely many symbols i.e., unlike earlier numerical systems, in which only finitely many natural num- bers could be expressed using finitely many symbols. This revolutionary importance of the development of place-value decimal numerals in India was recognized, for in- stance, in writings of the 18-th century French mathematician Laplace. In this context, it is also of interest to observe that the discussion in (ii) of the interpreta- tion of the discussion of (i) in terms of line segments is reminiscent of the discussion of the “Euclidean algorithm” in Euclid’s Elements, in which numbers are often rep- resented as lengths of line segments. Finally, we recall that the Boolean aspects discussed in (iii) played an important role in the [well-known!] development of modern digital computers in the 20-th century. The gluings of adjacent line segments discussed in Examples 2.3.2, 2.4.1, 2.4.2 may in some sense be regarded as a sort of optimized elementary geometric/combinatorial representation of the essential logical “∧/∨” structure surrounding a gluing in a fashion that is qualitatively entirely structually similar to the gluings that occur in inter-universal Teichmüller theory, which will be discussed in more detail in §3 below. The somewhat more numerical/arithmetic situations discussed in Examples 2.4.3, 2.4.5, 2.4.6 may also be regarded as entirely elementary representations of this essential logical “∧/∨” structure surrounding a gluing. On the other hand, the glu- ing operation that occurs in the standard construction of the projective line, while somewhat less elementary than the previously mentioned examples, also constitutes an important and, moreover, still relatively elementary! representation of this essential logical “∧/∨” structure surrounding a gluing. Moreover, this example of the projective line discussed in Example 2.4.7 is more directly related to scheme- theoretic arithmetic geometry than the previously mentioned examples and 58 SHINICHI MOCHIZUKI helps to motivate the subsequent ring-/monoid-theoretic Example 2.4.8, which may literally be regarded, i.e., in a much more rigorous, technical sense, as a sort of miniature qualitative model that is to say, so to speak, a sort of “preview” of the gluing constituted by the Θ-link of inter-universal Teichmüller theory. Example 2.4.7: The projective line as a gluing of ring schemes along a multiplicative group scheme. In the following discussion, we take k to be a field def and q k to be an element such that q 3  = q [i.e., q ∈ {0, 1, −1}]. Write k × = k\{0}, A 1 for the affine line Spec(k[T ]) over k, G m for the open subscheme Spec(k[T, T −1 ]) of A 1 obtained by removing the origin. Thus, the standard coordinate T on A 1 , G m determines natural bijections A 1 (k) k, G m (k) k × of the respective sets of k-rational points of A 1 , G m with corresponding subsets of k. Also, we recall that A 1 is equipped with a well-known natural structure of ring scheme over k, while G m is equipped with a well-known natural structure of [multiplicative] group scheme over k. (i) Write A 1 , A 1 for the k-ring schemes given by copies of A 1 equipped with the respective labels “†”, “‡”. We regard A 1 as being further equipped with the k-rational point q −1 A 1 (k) ( k) corresponding to the multiplicative inverse of the element q k and A 1 as being further equipped with the k-rational point q A 1 (k) ( k) corresponding to the element q k. Similarly, we write G m , G m for the [multiplicative] k-group schemes given by copies of G m , equipped with the respective labels “†”, “‡”. Thus, q −1 G m (k) (⊆ A 1 (k)), q G m (k) (⊆ 1 A (k)). (ii) Relative to the notation of (i), we observe that (ii-a) there exists a unique isomorphism of k-ring schemes A 1 A 1 , but that (ii-b) the pairs ( A 1 , q −1 ) and ( A 1 , q) are not isomorphic, i.e., as pairs consisting of a k-ring scheme equipped with a k-rational point [cf. our assumption that q 3  = q]. By contrast, (ii-c) there exists a unique isomorphism of pairs ( G m , q −1 ) ( G m , q), i.e., of pairs consisting of a [multiplicative] k-group scheme equipped with a k-rational point. Here, we observe that the isomorphism ( G m , q −1 ) ( G m , q) of (ii-c) does not extend [cf. (ii-b)!] to an isomorphism ( A 1 , q −1 ) ( A 1 , q). In particular, (ii-d) the isomorphism of [multiplicative] k-group schemes G m G m is not compatible with the k-ring scheme structures of A 1 (⊇ G m ), 1 A (⊇ G m ). Next, we observe that (ii-e) the standard construction of the projective line may be regarded as the result of gluing ( A 1 , q −1 ) to ( A 1 , q) along the isomorphism ( G m , q −1 ) ( G m , q) LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 59 of (ii-c); thus, relative to this gluing, G m G m may be regarded si- multaneously as an open subscheme of A 1 AND [cf. “∧”!] as an open subscheme of A 1 . In particular, (ii-d), (ii-e) may be summarized as follows: the standard construction of the projective line may be regarded as a gluing of two ring schemes along an isomorphism of multiplicative group schemes that is not compatible with the ring scheme struc- tures on either side of the gluing. Moreover, we note that, relative to this gluing, (ii-f) the notion of a regular function on A 1 cannot be expressed di- rectly in terms of the notion of a regular function on A 1 , whereas (ii-g) the notion of a rational function on A 1 can be expressed directly in terms of i.e., in essence, coincides, relative to the above gluing, with the notion of a rational function on A 1 . Finally, we observe that (ii-h) if, in the gluing of (ii-e), one arbitrarily deletes the distinct labels “†”, “‡”, then the resulting “gluing without labels” amounts to a gluing of a single copy of A 1 to itself that maps the standard coordinate “T on A 1 [regarded, say, as a rational function on A 1 ] to T −1 ; that is to say, such a “gluing without labels” results in a contradiction [i.e., since T  = T −1 !], unless one passes to some sort of quotient of A 1 which amounts, from a foundational/logical point of view, to the introduction of some sort of indeterminacy, i.e., to the consideration of some sort of collection of possibilities [cf. “∨”!]. (iii) The discussion of the projective line in (ii) is truly remarkable in that it completely parallels i.e., relative to the correspondence “−1” ←→ “j 2 between the exponent “−1” in the discussion of (ii) and the exponents “j 2 ”, where j ranges from 1 to l  , in the discussion of §3.4 numerous aspects of the Θ-link of inter-universal Teichmüller theory, which we shall discuss in more detail in §3 [cf., especially, §3.4]. Indeed, (iii-a) the isomorphism of (ii-a) may be understood as corresponding to the fact that the ±ell NF-)Hodge theaters on either side of the Θ-link in inter-universal Teichmüller theory are isomorphic, while (iii-b) the observation of (ii-b) may be understood as corresponding to the fact that there is no isomorphism of ±ell NF-)Hodge theaters as in (iii- a) that maps the Θ-pilot in the domain of the Θ-link [which corresponds to q −1 ”] to the q-pilot in the codomain of the Θ-link [which corresponds to q”]. On the other hand, 60 SHINICHI MOCHIZUKI (iii-c) the isomorphism of (ii-c) may be understood as corresponding to the full poly-isomorphism of [multiplicative!] F ×μ -prime-strips that constitutes the Θ-link, while (iii-d) the observation of (ii-d) may be understood as corresponding to the fact that the full poly-isomorphism of (iii-c) is not compatible with the ring structures determined by the ±ell NF-)Hodge theaters on either side of the Θ-link, i.e., in particular, does not arise from a poly-isomorphism between these ±ell NF-)Hodge theaters on either side of the Θ-link [cf. (iii-b)]. Next, we observe that (iii-e) the gluing of (ii-e) may be understood as corresponding to the glu- ing constituted by the Θ-link between the ±ell NF-)Hodge theaters on either side of the Θ-link, i.e., a gluing along [multiplicative!] F ×μ - prime-strips that is not compatible with the ring structures in the domain and codomain of the Θ-link, but which allows one to obtain a single F ×μ -prime-strip, up to isomorphism, that may be interpreted simultaneously as the F ×μ -prime-strip arising from the Θ-pilot in the domain of the Θ-link AND [cf. “∧”!] as the F ×μ -prime-strip arising from the q-pilot in the codomain of the Θ-link. Here, we recall that this crucial logical AND “∧” property of the Θ-link is the central theme of the present paper [cf. the discussion of Examples 2.4.1, 2.4.2, 2.4.3, 2.4.4, 2.4.5, 2.4.6!]. Next, we observe that (iii-f) the observation of (ii-f) may be understood as corresponding to the fact that, at least from an a priori point of view, there is no natural way to express the Θ-pilot of the ±ell NF-)Hodge theater in the domain of the Θ-link, relative to the gluing of (iii-e), in terms of the ±ell NF-)Hodge theater in the codomain of the Θ-link, while (iii-g) the observation of (ii-g) may be understood as corresponding to the si- multaneous holomorphic expressibility (SHE) property of the mul- tiradial representation of the Θ-pilot [cf. [IUTchIII], Remark 3.11.1, (iii); [IUTchIII], Remark 3.9.5, (viii), (ix)], which allows one to express the Θ-pilot of the ±ell NF-)Hodge theater in the domain of the Θ-link, relative to the gluing of (iii-e), in terms of the ±ell NF-)Hodge theater in the codomain of the Θ-link [cf. also the discussion of (iv), (v), below; the discussion surrounding Example 2.4.8, (iii-a), (iii-b), below]. Finally, we note that (iii-h) the “gluing without labels” discussed in (ii-h) may be understood as cor- responding to the oversimplified version “RCS-IUT” of inter-universal Teichmüller theory obtained by implementing the RCS-identifications of RCS-redundant copies of objects [cf. the discussion of §1.2, Example 2.4.5], which leads to an immediate contradiction, unless one introduces some sort of quotient/indeterminacy, i.e., which amounts to the con- sideration of some sort of collection of possibilities [cf. “∨”!]. In particular, relative to this remarkably close structural resemblance between the gluing that appears in the standard construction of the projective line and LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 61 the gluing constituted by the Θ-link of inter-universal Teichmüller theory, the central assertion “IUT RCS-IUT” of the RCS [cf. the discussion of Example 2.4.5] may be understood as correspond- ing to the assertion of an “obvious equivalence” [cf. the discussion of §1.3] between · the projective line, on the one hand, and · the affine line regarded up to some sort of identification of the standard coordinate “T on the affine line with its inverse, on the other. (iv) From the point of view of the analogy discussed in (iii) between the gluing construction of the projective line and inter-universal Teichmüller theory, perhaps the closest nontrivial analogue, in the case of the projective line, to the multiradial representation of the Θ-pilot in inter-universal Teichmüller theory is the group of projective general linear [i.e., “P GL 2 ”] symmetries of the projective line [cf. also the discussion of (v) below; the discussion of Example 3.10.1, (i), below]. That is to say, although there is also an analogy, discussed in (iii-g), with the observation of (ii-g), the content of (ii-g) is rather formal/trivial. By contrast, the P GL 2 -symmetries of the projective line are somewhat less trivial, especially from the point of view of the gluing construction of the projective line discussed in (ii) [cf. also the discussion of (v) below; the discussion of Example 3.10.1, (i), below]. (v) The P GL 2 -symmetries of the projective line are, in some sense, especially interesting in the case where one takes k to be the field C of complex numbers, and one restricts to the subgroup P U 2 P GL 2 (C) given by the image of the unitary matrices, i.e., the projective unitary group. Thus, as is well-known, one may think of P U 2 as the group of isometric symmetries of the Riemann surface associated to the projective line over C equipped with the Fubini- Study metric. The underlying topological space of this Riemann surface may be naturally identified with the sphere S 2 . The geodesics associated to the Fubini-Study metric then correspond to great circles on the sphere S 2 [cf., e.g., the illustration of [GeoSph]]. In particular, the geodesics that pass through the north/south poles of S 2 may be thought of as lines of longitude. In the current metrized situation, it is natural to think of S 2 as being obtained not via a gluing of the complement of the north pole to the complement of the south pole [i.e., as in (ii-e)], but rather as being obtained via a gluing S 2 H + E = H + H def H S 2 of the northern hemisphere H + S 2 of S 2 to the southern hemisphere H S 2 of S 2 along the equator E S 2 . Here, we note that the gluing of (ii-e) which yields a single rational function on the projective line that corresponds simultaneously to the standard coordinate T on A 1 AND to the standard coordinate T −1 on 1 A may be thought of, in the current metrized situation, as corresponding to the following [at first glance, self-contradictory!] phenomenon: 62 SHINICHI MOCHIZUKI (OrFlw) an oriented flow along the equator which may be thought of phys- ically as a sort of wind current that flows from east to west appears simultaneously to be flowing, from the point of view of the northern hemi- sphere H + S 2 , in the clockwise direction AND, from the point of view of the southern hemisphere H S 2 , in the counterclockwise direction. Next, let us recall that unlike S 2 ! both H + and H may be thought of as closed discs in the plane. Thus, in summary, (GdsFlw) the geodesic geometry of the Fubini-Study metric i.e., in essence, the (P GL 2 (C) ⊇) P U 2 -symmetries of S 2 allow one, by considering the geodesic flow along lines of longitude, to represent, up to a relatively mild distortion, the entirety of S 2 , i.e., including H S 2 , as a sort of extension/deformation of the closed disc H + . Indeed, (GdsFlw) is precisely the principle that is applied to represent, using lines of longitude, the globe [i.e., in the sense of the surface of the planet earth] via a rectangular, planar, cartesian map [i.e., in the sense of cartography]! Note, moreover, that (NoLbDlt) although the approach of (GdsFlw) gives rise to a certain relatively mild degree of distortion in the representation of H in terms of H + , it does not involve any sort of naive identification of the closed discs H + , H , i.e., any sort of arbitrary label deletion, in the style of (ii-h). The interpretation discussed in (GdsFlw) and (NoLbDlt) of the (P GL 2 (C) ⊇) P U 2 - symmetries of S 2 may be understood as strongly suggesting a nontrivial analogy between these symmetries of S 2 and the multiradial representation of the Θ- pilot in inter-universal Teichmüller theory [cf. the analogy between multiradiality and connections/parallel transport/crystals discussed in [Alien], §3.1, (iv), (v), as well as §3.5, §3.10, below]. Example 2.4.8: Gluings of rings along multiplicative monoids. (i) Let R be an integral domain equipped with the action of a group G and N a positive integer 2. For simplicity, we assume that N = 1 + · · · + 1 [i.e., the sum of N copies of “1 R”] determines a nonzero element of R. Write · R  R for the multiplicative monoid of nonzero elements of R; · R   R for the quotient multiplicative monoid of R  by the group of roots of unity of R; · (R  ) N R  , (R ) N R for the multiplicative submonoids consist- ing of the N -th powers of elements of “(−)”. Thus, G acts naturally and in a compatible fashion not only on the ring R, but also on the multiplicative monoids R  , R , (R  ) N , (R ) N . Also, we observe that the N -th power map on R determines an isomorphism of multiplicative monoids equipped with actions by G R (R ) N (⊆ R ) that does not arise from a ring homomorphism, i.e., as may be seen from the fact that this isomorphism of multiplicative monoids is not compatible with the LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 63 operation of addition [cf. our assumption that N determines a nonzero element of R!] . (ii) Let R, R be two distinct copies of the integral domain R of (i), equipped with respective actions by two distinct copies G, G of the group G of (i). We shall use similar notation for objects labeled with “†” or “‡” to the notation introduced in (i) for objects not equipped with such labels. Then (ii-a) one may use the isomorphism of multiplicative monoids arising from the N -th power map discussed in (i) to glue together G  R ( R  ) N  ( R ) N R  R  R  G the ring R to the ring R along the multiplicative monoid ( R ) N R [cf. the discussion of Example 2.4.7, (ii), (iii)!]. Here, we observe that this gluing is compatible with the respective actions of G, G relative to the isomorphism G G given by forgetting the labels “†”, “‡”, but in this context, it is of the utmost importance to remember that (ii-b) since, as observed in (i), the N -th power map is not compatible with the operation of addition (!), this isomorphism G G may be regarded either as an isomorphism of abstract groups or as an isomorphism of groups equipped with actions on certain multiplicative monoids, but not as an isomorphism of groups equipped with actions on rings [i.e., R, R], e.g., as is the case where G, G are taken to be “Galois groups” [that is to say, groups equipped with faithful actions on some field, such as the quotient field of R or R]. In the context of (ii-b), we observe that, of course, one may also consider taking the point of view that G, G are groups equipped with actions on the diagram R ( R  ) N  ( R ) N R  R  R [consisting of various rings, multiplicative monoids, etc.] of (ii-a), i.e., not just on some isolated portion of the diagram such as R, ( R ) N , R , or R. (ii-c) The fundamental and indeed essentially tautological! problem, however, with this approach of thinking of G, G as groups of automor- phisms of the diagram of the above display is that this approach yields a situation in which one can no longer consider [i.e., in the sense that it is no longer a well-defined proposition to consider!] various isolated portions of the diagram [i.e., such as R, ( R  ) N , R , or R] equipped with actions by G, G independently of the entire diagram. On the other hand, as we shall see in (iii) below, the main issue of interest sur- rounding the gluing of (ii-a) involves consideration of the extent to which one can start precisely from such an isolated portion of the diagram namely, the glued data G  ( R ) N R  G and then proceed to reconstruct, possibly up to relatively mild indeterminacies, some remaining portion of the diagram. Finally, we observe that the importance, in 64 SHINICHI MOCHIZUKI the context of inter-universal Teichmüller theory, of thinking of Galois groups [not as groups of automorphisms of ring/fields/diagrams involving rings (!), but rather] as abstract groups [i.e., as emphasized in the above discussion!] is reminiscent of the discussion of the issue of the “relative subordination” of group theory versus field theory [i.e., “group theory field theory” versus “field theory group theory”] in [Alien], §4.4, (i). (iii) In general, in the situation of the gluing considered in (ii-a), (iii-a) the problem of describing the additive structure of R in terms of the additive structure of R in a fashion that is compatible with the gluing and via a single algorithm that may be applied to the glued data to reconstruct simultaneously the additive structures of both R and R seems to be hopelessly intractable! The nontriviality of this problem may already be seen, for instance, in the case where one takes R to be Z [i.e., the ring of rational integers]. Indeed, this sort of problem may be understood as (iii-b) the starting point of inter-universal Teichmüller theory, where one consid- ers the gluing constituted by the Θ-link [cf. the discussion of §3.4 below] and the issue of describing in a fashion compatible with the crucial logical AND property [cf. the discussion of (iv) below] associated to this gluing! certain portions of the ring/additive structure of the domain [i.e., labeled by “†”] of the Θ-link in terms of the ring/additive structure of the codomain [i.e., labeled by “‡”] of the Θ-link via a sin- gle algorithm that may be applied to the glued data to reconstruct si- multaneously the corresponding portions of the ring/additive structure of both the domain and the codomain of the Θ-link [cf. the discussion of the simultaneous holomorphic expressibility (SHE) property in [IUTchIII], Remark 3.11.1, (iii); [Alien], §3.7, (i); [Alien], §3.11, (iv)]. Such a description is ultimately achieved in inter-universal Teichmüller theory by means of the multiradial representation of the Θ-pilot, which allows one to reconstruct, up to relatively mild indeterminacies, certain portions of interest of the ring/additive structure of the domain of the Θ-link in terms of the ring/additive structure of the codomain of the Θ-link [cf. the discussion of Example 2.4.7, (v)] in a fashion that is compatible with the gluing and via a single algorithm that may be applied to the glued data to reconstruct simultaneously the corresponding portions of the ring/additive structure of both the domain and the codomain of the Θ-link by making use of certain structural properties of the various multiplicative monoids equipped with group actions that appear in the construction of the Θ-link, as well as certain highly nontrivial anabelian properties of the underlying abstract groups of the various Galois groups that appear [cf. the discussion of (ii-b) above; the discussion of §3.2, §3.8, below]. In this context, it is also interesting to note that, when N = p is a prime number, the fact that the Frobenius morphism given by raising to the power p is a ring homomorphism in characteristic p may be interpreted in the following way: (iii-c) even in the situation of the present discussion [i.e., where the ring R is not of positive characteristic!], the isomorphism of multiplicative monoids LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 65 obtained by raising to the p-th power i.e., the isomorphism of mul- tiplicative monoids that appears in the gluing of (ii-a) may in fact be regarded as being “simultaneously compatible” with the additive structures in its domain and codomain if one regards one’s computations as being subject to the “indeterminacy” given by working modulo p. Finally, we observe that this interpretation is reminiscent of the important analogies between inter-universal Teichmüller theory, on the one hand, and Frobenius liftings and p-adic Teichmüller theory, on the other, as discussed in [Alien], §2.4, §2.5; [Alien], §3.3, (ii) [cf. also the discussion of crystals in [Alien], §3.1, (v), as well as §3.5, §3.10 below]. (iv) In the context of the gluing of (ii-a), we observe that (iv-a) the glued multiplicative monoid ( R ) N R , regarded up to iso- morphism, is simultaneously · the multiplicative monoid ( R ) N associated to the ring R AND · the multiplicative monoid R associated to the ring R. In a similar vein, if one thinks of the glued group G G, regarded up to iso- morphism, either as an abstract group or as a group equipped with an action on the glued multiplicative monoid, then (iv-b) this glued group G G is simultaneously · the group G equipped with an action on the multiplicative monoid ( R ) N associated to the ring R AND · the group G equipped with an action on the multiplicative monoid ( R ) N associated to the ring R. The gluing given by the Θ-link involves entirely analogous logical AND prop- erties, which are fundamental to the essential logical structure of inter-universal Teichmüller theory, as exposed in the present paper. Of course, (iv-c) it is always possible to consider the situation in which one deletes the labels “†”, “‡”, but only at the expense of sacrificing these crucial log- ical AND properties (iv-a), (iv-b), i.e., at the expense of agreeing to work under the assumption that the “glued data” is the data associated to “†” OR the data associated to “‡”, but not necessarily both simultaneously. Moreover, once one deletes the labels “†”, “‡” i.e., so that the two copies of “R” are identified with one another via an isomorphism of rings! the problem of describing the ring/additive structure of one copy in terms of the ring/additive structure of the other copy [cf. the discussion of (iii)!] becomes “trivial”, but this triviality is of little interest since it is achieved only at the cost of sacrificing the crucial logical AND properties in favor of the [entirely uninteresting!] logical 66 SHINICHI MOCHIZUKI OR property just described cf. the discussion surrounding (NeuORInd) in §3.4 below, as well as the discussion of IUT = AND-IUT versus RCS-IUT/OR-IUT/EssOR-IUT in Examples 2.4.5, (ii), (iii), (v), (vi), (vii); 2.4.7, (ii), (iii), (v). (v) Finally, we note that the relationship between the discussion of the present Example 2.4.8 and the numerical situation discussed in Example 2.4.5, (ii), (iii) [cf. also the discussion of the final page and a half of the files “[SS2018-05]”, “[SS2018- 08]” available at the website [Dsc2018]] may be seen by considering the case where R is taken to be the ring of rational integers Z [or, in fact, slightly more generally, the ring of integers “O F of a finite field extension F of the field of rational numbers Q a situation that may be related to the case of Z by applying the multiplicative norm map N F/Q : F Q]. Indeed, in the case where R is taken to be Z, one may consider the “height” log(|x|) R [where “log” denotes the natural logarithm of a positive real number, and “| |” denotes the absolute value of an element of Z] associated to a nonzero element 0  = x Z. Then the N -th power map of (i), (ii) corresponds, after passing to heights, to multiplying real numbers by N , i.e., in essence to the situation considered in Example 2.4.5, (ii), (iii) [which corresponds to the case where the “N of the present discussion is taken to be 2]. Section 3: The logical structure of inter-universal Teichmüller theory In the present §3, we give a detailed exposition of the essential logical struc- ture of inter-universal Teichmüller theory, with a special focus on issues related to RCS-redundancy. From a strictly rigorous point of view, this exposition assumes a substantial level of knowledge and understanding of the technical details of inter- universal Teichmüller theory [which are surveyed, for instance, in [Alien]]. On the other hand, in a certain qualitative sense, the discussion of the present §3 may in fact be understood, at a relatively elementary level, via the analogies that we dis- cuss with the topics covered in §2. Indeed, in this context, it should be emphasized that, despite the relatively novel nature of the set-up of inter-universal Teichmüller theory, the essential mathematical content that lies at the heart of all of the issues covered in the present §3 concerns entirely well-known mathematics at the advanced undergraduate or beginning graduate level [i.e., the topics covered in §2]. §3.1. One-dimensionality via identification of RCS-redundant copies Inter-universal Teichmüller theory concerns the explicit description of the re- lationship between various possible intertwinings namely, the “Θ”- and “q-” intertwinings LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 67 between the two underlying combinatorial/arithmetic dimensions of a ring [cf., e.g., [Alien], §2.11; [Alien], §3.11, (v), as well as the discussion of §3.9 below]. There are many different ways of thinking about these two underlying combinatorial/arithmetic dimensions of a ring; one way to understand these two dimensions is to think of them as corresponding, respectively, to the unit group and value group of the various local fields that appear as completions of a number field at one of its valuations. In more technical language, this sort of decomposition into unit groups and value groups may be seen in the F ×μ -prime-strips that appear in the Θ-link of inter-universal Teichmüller theory. Thus, if one thinks in terms of such F ×μ - prime-strips, then inter-universal Teichmüller theory may be summarized as follows: (2-Dim) The main content of inter-universal Teichmüller theory consists of an explicit description, up to certain relatively mild indeterminacies, of the Θ-intertwining on the [two-dimensional!] F ×μ -prime-strips that appear in the Θ-link in terms of the q-intertwining on these F ×μ - prime-strips by means of the log-link and various types of Kummer theory that are used to relate Frobenius-like and étale-like structures [cf. the discussion of Example 2.4.8, (iii)]. In particular, the essential mathematical content of inter-universal Teichmüller the- ory concerns an a priori variable relationship between the two underlying com- binatorial/arithmetic dimensions of a ring. Put another way, if one arbitrarily “crushes” these two dimensions into a single dimension i.e., in more technical language, assumes that (1-Dim) there exists a consistent choice of a fixed relationship between these two dimensions of (2-Dim), so that these two dimensions may, in effect, be regarded as a single dimension then one immediately obtains a superficial contradiction [cf. the discussion of Example 3.1.1, (i-b), (ii-b), below]. Indeed, this is one of the central assertions of the RCS [cf. the discussion following Example 3.1.1]. This is not a “new” observation, but rather, in some sense, the starting point of inter-universal Teichmüller theory, i.e., the initial motivation for regarding the relationship between the two underlying combinatorial/arithmetic dimensions of a ring as being variable, rather than fixed. Example 3.1.1: Elementary models of gluings and intertwinings. In the following, we shall write V for the topological group R >0 . Let x, y V be [not necessarily distinct!] elements of V and Y V a nonempty subset of V . Let V rl , V im be two distinct labeled copies of V , which we think of as corresponding to the positive portions of the real and imaginary axes in the complex plane. (i) Let V , V be two not necessarily distinct copies of V . We shall write y V , x V for the respective elements determined by y, x V . (i-a) Suppose that V , V are distinct copies of V . Write W for the topolog- ical space obtained by gluing V , V along the homeomorphic subspaces { y} V , { x} V . Then observe that this construction of W is well- defined and free of any internal contradictions. Moreover, the existence of W does not imply any nontrivial conclusions concerning x and y. 68 SHINICHI MOCHIZUKI Note the sharp contrast between the situation discussed in (i-a) and the following situation: def (i-b) Suppose that V , V are in fact the same copy of V , i.e., V = V = V . Consider the assertion that the topological space V is obtained by gluing V , V along the homeomorphic subspaces { y} V , { x} V . Then observe that this assertion concerning V is well-defined and free of internal contradictions only in the case where x = y. That is to say, the existence of a topological space V as described in the above assertion implies the nontrivial conclusion that x = y, or, equivalently, a “con- tradiction” to the assertion that x  = y. One may also consider the following variant of (i-b): (i-c) One replaces { y} V in (i-b) by the nonempty subset Y V [i.e., determined by Y V ], where one thinks of this subset as a set of “possible y’s”. The resulting “assertion” then becomes a corresponding collection of assertions related by logical OR “∨’s”, and the final nontrivial con- clusion is that x Y . (ii) The elementary models presented in (i) may be interpreted as essentially equivalent representations of various models of “holomorphic structures” [cf. the discussion below of (InfH), as well as Examples 3.3.1, 3.3.2] i.e., in the ter- minology of the discussion preceding the present Example 3.1.1, “intertwinings” between the “real” and “imaginary” dimensions V rl , V im . Here, we think of “holomorphic structures”/“intertwinings” as being defined by assignments V rl  1 rl ? V im [where 1 rl V rl denotes the element determined by 1 V ], corresponding to “coun- terclockwise rotations by 90 degrees”, or, alternatively, “multiplication by −1”. In- deed, let V rl , V rl be two not necessarily distinct copies of V rl ; V im , V im two not necessarily distinct copies of V im . We shall write y im V im , x im V im for the respective elements determined by y, x V . Then the discussion of (i-a) may be translated into a discussion concerning intertwinings by arguing as follows: (ii-a) Suppose that V rl , V rl are distinct copies of V rl ; V im , V im are distinct copies of V im . Here, we think of V rl , V im as being equipped with the intertwining given by taking “?” to be y im V im ; we think of rl im V , V as being equipped with the intertwining given by taking “?” to be x im V im . Then one applies (i-a), relative to the correspondences im V ←→ V , V im ←→ V . This yields a gluing as in (i-a) that is well- defined and free of any internal contradictions. Moreover, the existence of such a gluing does not imply any nontrivial conclusions concerning x and y. In a similar vein: def (ii-b) Suppose that V rl , V rl are in fact the same copy of V rl , i.e., V rl = rl V = V rl , and that V im , V im are in fact the same copy of V im , i.e., LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 69 def V im = V im = V im . Then one applies (i-b), relative to the correspon- dence V im ←→ V . This yields an assertion concerning a gluing as in (i-b) i.e., in the language of the present discussion, concerning a coin- cidence of the intertwining on V rl , V im with the intertwining on V rl , im V that is well-defined and free of internal contradictions only in the case where x = y. That is to say, the existence of such a gluing implies the nontrivial conclusion that x = y, or, equivalently, a “contradiction” to the assertion that x  = y. (ii-c) One replaces { y im } V im in (ii-b) [cf. also the notation of (ii-a)] by the nonempty subset Y im V im [i.e., the subset determined by Y V ], where one thinks of this subset as a set of “possible y’s”. The resulting “assertion” then becomes a corresponding collection of assertions related by logical OR “∨’s”, and the final nontrivial conclusion is that x Y . (iii) Relative to the analogy with inter-universal Teichmüller theory, we have correspondences with objects that appear in the elementary models of (ii) as follows: V rl ←→ the value group portion of an F ×μ -prime-strip; V im ←→ the unit group portion of an F ×μ -prime-strip; †/‡ ←→ ±ell NF-)Hodge theaters in the domain/codomain of the Θ-link; intertwinings involving “y” ←→ the Θ-intertwinings; intertwinings involving “x” ←→ the q-intertwinings; [cf. the discussion at the beginning of §3.4]. Here, we note that from the point of view of intertwinings, the unit group portion corresponding to “V im must be understood as being log-shifted by −1, relative to the value group portion corre- sponding to “V rl [cf. the discussion below of (InfH), as well as Examples 3.3.1, 3.3.2]. That is to say, if the value group portion corresponding to “V rl is located at the coordinate (n, m) of the log-theta-lattice, then the unit group portion corre- sponding to “V im must be understood as being located at the coordinate (n, m−1) of the log-theta-lattice. In particular, the unit group and value group portions cor- responding to a pair “(V rl , V im )” belong to different F ×μ -prime-strips. From the point of view of the discussion of (1-Dim), the “consistent choice of a fixed relationship” corresponds to the coincidence of intertwinings in (ii-b), while the resulting “superficial contradiction” corresponds to the “contradiction” discussed in (ii-b). On the other hand, the “explicit description”/“variable relationship” of (2-Dim), which leads naturally to a numerical estimate/inequality concerning log- volumes [cf. Example 2.4.5, (ii)], corresponds to the situation involving various possibilities discussed in (ii-c), which leads to the nontrivial conclusion “x Y [cf. the discussion of “closed loops” in (Stp7), (Stp8) of §3.10 below; the discus- sion of (DltLb) in §3.11 below; the discussion of [IUTchIII], Remark 3.12.2, (ii)]. (iv) Finally, we observe in passing that the fixed intertwining of (ii-b) [cf. also the discussion of (ii-b) in (iii), as well as the discussion of (FxRng), (FxEuc), (FxFld), (RdVar) below] may be regarded as being analogous to the well-known classical holomorphic approach to the theory of moduli of [one-dimensional] 70 SHINICHI MOCHIZUKI complex tori, that is to say, in which one works with a copy of the upper half- plane “H” with a fixed holomorphic structure and thinks of the moduli of complex tori as a “variation of period matrices” [i.e., the holomorphic parameter “z H”, which may be taken, in the notation of (ii-b), to be “ix” or “iy”]. By contrast, the situation involving the set Y im V im discussed in (ii-c) may be regarded as analogous to the [real analytic] Teichmüller approach to the theory of moduli of complex tori [cf. the discussion of Example 3.3.1], i.e., in which the holomorphic structure is subject to Teichmüller dilations [corresponding to various elements in the set Y im ], relative to the fixed “real analytic” pair given by V rl , V im . One central assertion of the RCS which appears, for instance, in certain 10pp. manuscripts written by adherents of the RCS [cf., especially, the discussion of the final page and a half of the files “[SS2018-05]”, “[SS2018-08]” available at the website [Dsc2018]] is to the effect that the existence, as in (1-Dim), of a consistent choice of a fixed relationship between the two dimensions of (2-Dim) may be derived as a consequence i.e., in more succinct notation, (RC-FrÉt), (RC-log), (RC-Θ) =⇒ (1-Dim) of certain “redundant copies assertions”, as follows: (RC-FrÉt) the Frobenius-like and étale-like versions of objects in inter-universal Teichmüller theory are “redundant”, i.e., may be identified with one another without affecting the essential logical structure of the theory; (RC-log) the ±ell NF-)Hodge theaters on either side of the log-link in inter- universal Teichmüller theory are “redundant”, i.e., may be identified with one another without affecting the essential logical structure of the theory; (RC-Θ) the ±ell NF-)Hodge theaters on either side of the Θ-link in inter- universal Teichmüller theory are “redundant”, i.e., may be identified with one another without affecting the essential logical structure of the theory. In the remainder of the present §3 [cf., especially, §3.2, §3.3, §3.4], we discuss in more detail the falsity of each of these “RCS-redundancy” assertions [i.e., (RC-FrÉt), (RC-log), (RC-Θ)]. Here, it should be noted that this falsity of (RC-FrÉt), (RC-log), (RC-Θ) is by no means a difficult or subtle issue, but rather a sort of matter of “belaboring the intuitively obvious” from the point of view of mathematicians who are thor- oughly familiar with inter-universal Teichmüller theory. Nevertheless, as discussed in [Rpt2018], §17, it is a pedagogically meaningful exercise to write out and discuss the details surrounding this sort of issue. Moreover, as discussed in §1.5 of the present paper, it is desirable from a historical point of view to produce detailed, explicit, and readily accessible written expositions concerning this sort of issue. This state of affairs prompts the following question: Why do adherents of the RCS continue to insist on asserting the validity of these assertions (RC-FrÉt), (RC-log), (RC-Θ)? LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 71 Any sort of complete, definitive answer to this question lies beyond the scope of the present paper. On the other hand, it seems natural to conjecture that one fundamental motivation for these assertions of RCS-redundancy may be found in the fact that (FxRng) many arithmetic geometers have only experienced working in situations where all schemes or, alternatively, rings that appear in a theory are regarded as belonging to a single category that is fixed throughout the theory, hence are related to another via ring homomorphisms, i.e., in such a way that the ring structure of the various rings involved is always respected [cf. the discussion of §1.5, as well as the discussion of §3.8 below]. It is not difficult to imagine that the heuristics and intuition that result from years [or decades!] of immersive experience in such mathematical situations could create a mindset that is fertile ground for the RCS-redundancy assertions that will be discussed in detail in the remainder of the present §3 [cf., especially, §3.2, §3.3, §3.4]. Finally, we observe that this situation is, in certain respects, reminiscent of various situations that occurred throughout the history of mathematics, such as, for instance, the situation that occurred in the late 19-th century with regard to such novel [i.e., at the time] notions as the notion of an abstract manifold or an abstract Riemann surface. That is to say, (FxEuc) from the point of view of anyone for whom it is a “matter of course” or “common sense” that all geometry must take place within some fixed, static ambient Euclidean space such as, for instance, the complex plane such abstract geometric notions as the notion of an abstract manifold or abstract Riemann surface might come across as deeply dis- turbing and unlikely to be of use in any substantive mathematical sense [cf. the discussion of §1.5; the discussion of [IUTchI], §I2]. In this context, it is of interest especially from a historical point of view to recall that, in some sense, the most fundamental classical example of such an abstract geometry is the Riemann surface that arises by applying the technique of analytic continuation to the complex logarithm, i.e., which may be regarded as a sort of distant ancestor [cf. the discussion of [IUTchI], Remark 5.1.4; [Alien], §3.3, (vi)] of the log-link of inter-universal Teichmüller theory. Another [in fact closely related!] fundamental classical example of such an abstract geometry is the hyperbolic geometry of the upper half-plane, which may also be regarded as a sort of distant ancestor of numerous aspects of inter-universal Teichmüller theory [cf. (InfH) and Example 3.3.2 in §3.3 below, as well as the discussion of [IUTchI], Remark 6.12.3, (iii); [IUTchIII], Remark 2.3.3, (ix), (x)]. Another historically important instance of this sort of situation may be seen in the introduction, in the early 19-th century, of Galois groups i.e., of [finite] automorphism groups of abstract fields as a tool for investigating the roots of polynomial equations. That is to say, (FxFld) until the advent of Galois groups/abstract fields, the issue of investigat- ing the roots of polynomial equations was always regarded again as a 72 SHINICHI MOCHIZUKI “matter of course” or “common sense” as an issue of investigating var- ious “exotic numbers” inside some fixed, static ambient field such as the field of complex numbers; moreover, from this more classical “common sense” point of view, the idea of working with automorphisms of abstract fields i.e., fields that are not constrained [since such constraints would rule out the existence of nontrivial automorphisms!] to be treated as sub- sets of some fixed, static ambient field might come across as deeply disturbing and unlikely to be of use in any substantive mathematical sense [cf. the discussion of §1.5]. On the other hand, from the point of view of inter-universal Teichmüller theory, this radical transition roots as concrete numbers  Galois groups/abstract fields that occurred in the early 19-th century may be regarded as a sort of distant ancestor of the transition Galois groups/abstract fields  abstract groups/anabelian algorithms that occurs in inter-universal Teichmüller theory [cf. the discussion at the beginning of §3.2 below; the discussion of §3.8 below; the discussion of the final portion of [Alien], §4.4, (i)]. A somewhat more recent historical example of this sort of situation may be seen in the situation surrounding the introduction of [possibly non-reduced] schemes by Grothendieck in the early 1960’s. Indeed, (RdVar) the possible existence of nilpotent elements in the structure sheaf of a non-reduced scheme struck many more classically oriented algebraic ge- ometers, who were accustomed to working only with reduced varieties whose geometry could be understood intuitively in terms of their sets of closed points as being entirely meaningless and unlikely to be of use in any substantive mathematical sense, especially since it was taken as a “matter of course” or “common sense” that any mathematically sub- stantive property of a variety would most certainly necessarily be readily identifiable at the level of the set of closed points of the variety [cf. the discussion of §1.5]. This more recent example of non-reduced schemes is especially of interest in the context of inter-universal Teichmüller theory in light of the strong structural re- semblances that exist between the notion of multiradiality in inter-universal Te- ichmüller theory and the theory [due to Grothendieck!] of crystals [cf. [Alien], §3.1, (v); the discussion of (CrAND) in §3.5 below; the discussion of §3.10 below]. Indeed, the “trivialization” of the theory of crystals that results from replacing the [non-reduced!] nilpotent thickenings that appear in the theory of crystals by the associated reduced schemes corresponds precisely to the situation discussed in (CrRCS) [cf. also (CrOR)] in §3.5 below. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 73 §3.2. RCS-redundancy of Frobenius-like/étale-like versions of objects We begin by recalling that ±ell NF-)Hodge theaters i.e., lattice points in the log-theta-lattice give rise to both Frobenius-like and étale-like objects. Whereas the datum of a Frobenius-like object depends, a priori, on the coordinates “(n, m)” of the ±ell NF-)Hodge theater from which it arises, étale-like objects satisfy various [horizontal/vertical] coricity properties to the effect that they map isomorphically to corresponding objects in a vertically [in the case of vertical coricity] or horizontally [in the case of horizontal coricity] neighboring ±ell NF- )Hodge theater of the log-theta-lattice [cf. the discussion of Example 3.2.2, (i), (iv), below; [Alien], §2.7, (i), (ii), (iii), (iv); [Alien], §2.8, 2 Fr/ét ; [Alien], §3.3, (ii), (vi), (vii); [Rpt2018], §15]. Here, we recall that étale-like objects correspond, for the most part, to arithmetic fundamental groups such as, for instance, the étale fundamental group “π 1 (X)” of a hyperbolic curve X over a number field or mixed characteristic local field or, more generally, to objects that may be reconstructed from such arithmetic fundamental groups, so long as the object is regarded as being equipped with auxil- iary data consisting of the arithmetic fundamental group from which it was recon- structed, together with the reconstruction algorithm that was applied to reconstruct the object. Here, we recall that, in this context, it is of fundamental importance that these arithmetic fundamental groups be treated simply as abstract topological groups [cf. the discussion of §3.8 below for more details]. Étale-like objects also satisfy a crucial symmetry property with respect to permutation of adjacent vertical lines of the log-theta-lattice [cf. Example 3.2.2, (ii), (iv), below; [Alien], §3.2; the discussion surrounding Fig. 3.12 in [Alien], §3.6, (i)]. That is to say, in summary, the crucial coricity/symmetry properties satisfied by étale-like objects which are, in essence, a formal consequence of treating the arithmetic fundamental groups that appear as abstract topological groups [cf. the discussion of §3.8 below for more details] play a central role in the multiradial algorithms of inter-universal Teichmüller theory [i.e., [IUTchIII], Theorem 3.11] and are not satisfied by Frobenius-like ob- jects cf., the discussion of Example 3.2.2, (i), (ii), (iv), below; [Alien], §2.7, (ii), (iii); [Alien], §3.1, (iii); [Alien], Example 3.2.2; [Rpt2018], §15, (LbΘ), (Lblog), (LbMn), (EtFr), (EtΘ), (Etlog), (EtMn). On the other hand, once one implements the RCS-identifications discussed in (RC-log), (RC-Θ), there is, in effect, “only one” ±ell NF-)Hodge theater in the log-theta-lattice, so all issues of determining relationships between corresponding objects in ±ell NF-)Hodge theaters at distinct coordinates “(n, m)” of the log- theta-lattice appear, at first glance, to have been “trivially resolved”. Put another way, 74 SHINICHI MOCHIZUKI once one implements the RCS-identifications of (RC-log), (RC-Θ), even Frobenius-like objects appear, at first glance, to satisfy all possible coric- ity/symmetry properties, i.e., at a more symbolic level, (RC-log), (RC-Θ) =⇒ (RC-FrÉt). In particular, the assertions of the RCS discussed in §3.1 and the present §3.2 may be summarized, at a symbolic level, as follows: (RC-log), (RC-Θ) =⇒ (RC-FrÉt), (RC-log), (RC-Θ) =⇒ (1-Dim). In fact, however, the RCS-identifications of (RC-log), (RC-Θ) do not resolve such issues [i.e., of relating corresponding objects in ±ell NF-)Hodge theaters at distinct coordinates “(n, m)” of the log-theta-lattice] at all [cf. the discussion of symmetries in Example 2.3.1, (iii)!], but rather merely have the effect of translating/reformulating such issues of relating corresponding objects in ±ell NF-)Hodge theaters at distinct coordinates “(n, m)” of the log-theta- lattice into issues of tracking the effect on objects in ±ell NF-)Hodge the- aters as one moves along the paths constituted by various composites of Θ- and log-links. On the other hand, at a purely formal level, the discussion given above of the falsity of (RC-FrÉt) i.e., as a conse- quence of the crucial coricity/symmetry properties discussed above is, in some sense, predicated on the falsity of (RC-log), (RC-Θ). This falsity of (RC-log), (RC-Θ) will be discussed in detail in §3.3, §3.4, below. In this context, it is useful to observe that the situation surrounding the Θ- link and (RC-Θ), (RC-FrÉt) (respectively, the log-link and (RC-log), (RC-FrÉt)) is structurally reminiscent of the object J discussed in Examples 2.3.2, 2.4.1, 2.4.2 [cf. also the correspondences discussed in Example 2.4.5, (ii); the discussion of [IUTchIII], Remark 1.2.2, (vi), (vii)], i.e., if one regards (StR1) the domain of the Θ- (respectively, log-) link as corresponding to I, (StR2) the codomain of the Θ- (respectively, log-) link as corresponding to I, (StR3) the gluing data i.e., a certain F ×μ -prime-strip (respectively, F- prime-strip) that arises from the domain ±ell NF-)Hodge theater of the Θ- (respectively, log-) link as corresponding to β = γ J , (StR4) the gluing data i.e., a certain F ×μ -prime-strip (respectively, F- prime-strip) that arises from the codomain ±ell NF-)Hodge theater of the Θ- (respectively, log-) link as corresponding to γ J = α, (StR5) the étale-like objects that are coric with respect to the Θ- (respectively, log-) link as corresponding to the glued differential discussed in (AOD3), and (StR6) the RCS-identification of (RC-Θ) (respectively, (RC-log)) as correspond- ing to the operation of passing to the quotient J  M = J/ I I L = I/ α β. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 75 This strong structural similarity will play an important role in the discussion of §3.3, §3.4, below. Finally, we observe that the portion, i.e., (StR5), of this strong structural similarity involving the glued differential discussed in (AOD3) is particularly of interest in the context of the discussion of [Alien], §2. That is to say, as discussed in the first paragraph of [Alien], §2.6, étale-like objects in inter-universal Teichmüller theory play an analogous role to the role played by tangent bundles/sheaves of differentials in the (InvHt) special case of the invariance of the height under isogenies between abelian varieties [due to Faltings] discussed in [Alien], §2.3, §2.4 [cf. also [Rpt2018], §16, (DiIsm), as well as the discussion of Example 3.2.1 below; the discussion of §3.5 below], as well as in the (FrDff) discussion of differentiation of p-adic liftings of the Frobenius mor- phism given in [Alien], §2.5. The efficacy of the technique of considering induced maps on differentials in the various examples discussed in [Alien], §2.3, §2.4, §2.5, may also be observed in the discussion of the fundamental theorem of calculus in §2.2, as well as in the context of (RC-FrÉt) and (StR5). In light of the central importance of (InvHt), as well as the closely related coricity/symmetry/commutativity properties of the log-theta-lattice [cf. the discussion at the beginning of the present §3.2, as well as the discussion at the beginning §3.3 below], in the essential logical structure of inter-universal Teichmüller theory, we pause to give a brief review/exposition of (InvHt) and these closely related coricity/symmetry/commutativity properties, in the following Examples 3.2.1, 3.2.2. Example 3.2.1: Global multiplicative subspaces and bounds on heights. (i) Let p be a prime number, K a finite extension of the field Q p of p-adic numbers, E an elliptic curve over K with bad multiplicative i.e., in other words, nonsmooth semi-stable reduction over the ring of integers O K of K. Write m K O K for the maximal ideal of O K . Thus, E is a Tate curve and hence [cf. the theory of [Mumf2]] may be represented, using the theory of formal schemes, as a sort of quotient Z “G m /q E of the multiplicative group scheme G m over K by the subgroup generated by the [nonzero] q-parameter q E m K of the elliptic curve E. Let l be a prime number. Then the subscheme μ l of l-torsion points of G m determines, via the above p-adic quotient representation, a canonical exact sequence 1 −→ μ l −→ E[l] −→ Z/lZ −→ 1 where we write E[l] E for the subgroup scheme of l-torsion points of E, and we observe that the generator “q E of the group of deck transformations of the above quotient representation determines a canonical generator γ l (E[l]/μ l )(K), up to multiplication by ±1, of (E[l]/μ l )(K), i.e., a canonical isomorphism, up to multiplication by ±1, of the quotient group scheme E[l]/μ l with [the group scheme over K determined by] Z/lZ. Thus, in summary, 76 SHINICHI MOCHIZUKI the subgroup scheme E[l] E of l-torsion points of E is equipped with a canonical multiplicative subspace μ l (→ E[l]), as well as with a canonical generator, up to multiplication by ±1, of the quotient E[l]/μ l . (ii) Let (E, 0 E ) be the pointed Riemann surface determined by an elliptic curve def over the field of complex numbers C. Write E[∞] = π 1 top (E) for the [usual topologi- cal] fundamental group of E, relative to the basepoint determined by the origin 0 E of the given elliptic curve. Thus, E[∞] is a free abelian group on two generators. Let M E[∞] be a rank one [free abelian] subgroup such that E[∞]/M is torsion-free. Then M corresponds to an infinite covering E M E of E such that any point 0 E M of E M that lifts 0 E determines, up to possible compo- sition with the inversion automorphism, an isomorphism E M C × of complex Lie groups, relative to the unique complex Lie group structure on the pointed Riemann surface (E M , 0 E M ) that lifts the unique complex Lie group structure on the pointed Riemann surface (E, 0 E ). In particular, we conclude that the choice of M [together with a choice of an isomorphism E M C × of the sort just discussed] determines a complex holomorphic quotient representation “C × /q E Z of the given elliptic curve, together with an exact sequence 1 −→ M −→ E[∞] −→ Z −→ 1 where we observe that the rank one free abelian group E[∞]/M ( = Z) is equipped with a unique choice of generator γ E[∞]/M, up to multiplication by ±1. Thus, the p-adic quotient representation, as well as the associated exact sequence and canonical generator [up to multiplication by ±1], discussed in (i) may be regarded as p-adic analogues of the complex holomorphic quotient representation and associated exact sequence/canonical generator [up to multiplication by ±1] discussed in the present (ii). (iii) Let F be a number field [i.e., a finite extension of the field Q of rational numbers] and E an elliptic curve over F . Write E[l] E for the subgroup scheme of l-torsion points of E. Then, in general, E does not necessarily admit a global multiplicative subspace (GMS) M E[l] or a global canonical generator (GCG) γ (E[l]/M )(F ), i.e., a subgroup scheme M E[l] or generator γ (E[l]/M )(F ) that restricts to the multiplicative subspace or canonical generator discussed in (i) at each nonarchimedean prime of F where E has bad multiplicative reduction. On the other hand, let us suppose, for the remainder of the present (iii), that E does admit a global multiplicative subspace (GMS) M E[l]. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 77 def Write E = E/M . Thus, we have an isogeny φ : E E that, in light of the discussion of (i), together with the fact that M E[l] is a GMS, corresponds to the isogeny Z l·Z −→ “G m /q E “G m /q E given by raising to the l-th power on G m at each nonarchimedean prime of F where E has bad multiplicative reduction. In particular, at each nonarchimedean prime v of F where E has bad multiplicative reduction, the q-parameter q E ,v of E at v is the l-th power of the q-parameter q E,v of E at v, i.e., l . q E ,v = q E,v In particular, if we write log(q (−) ) R for the normalized arithmetic degree “deg(−)” [cf. the discussion preceding [GenEll], Definition 1.2] of the arithmetic divisor determined by the q-parameters of an elliptic curve over a number field at the nonarchimedean primes where the elliptic curve, which we denote “(−)”, has bad multiplicative reduction, then we obtain the relation log(q E ) = l · log(q E ) R between log(q E ), log(q E ) R. (iv) We continue to consider the situation discussed in (iii). Write M ell M ell for the compactified moduli stack of elliptic curves or, equivalently, the moduli stack of pointed stable curves of type (1, 1) over Z and the open substack obtained by forming the complement of the divisor at infinity M ell M ell . Write ω M ell for the ample line bundle on M ell determined by the cotangent space at the origin of the tautological family of one-dimensional semi-abelian schemes over M ell [i.e., obtained by forming the complement of the unique node of the tautological pointed stable curve of type (1, 1) over M ell ]. Now recall the discriminant moduli form Δ M ell , which may be thought of as an isomorphism of line bundles O M ell ⊗12 ω M (−∞ M ell ) ell ⊗12 i.e., a section of ω M over M ell that has no zeroes or poles except for a zero of ell order 1 at M ell . It follows immediately from the existence of Δ M ell [cf., e.g., the discussion of [GenEll], §3, for more details] that, if, for the sake of simplicity, we ignore the contributions at the archimedean primes, then we obtain the relation ht (−) 1 6 log(q (−) ) where we write ht(−) for the normalized height associated to the ample line ⊗2 on M ell of the point determined by an elliptic curve “(−)” defined bundle ω M ell 78 SHINICHI MOCHIZUKI over a number field and “≈” to signify a relationship of bounded discrepancy [i.e., that the absolute value of the difference between the left- and right-hand sides is bounded by some positive real number independently of “(−)”]. (v) We continue to consider the situation discussed in (iii), (iv). Recall the isogeny φ : E E discussed in (iii). Since this isogeny is of degree l, hence, in particular, étale over nonarchimedean primes of F of residue characteristic  = l, we conclude immediately, via a straightforward computation of the map induced on differentials by φ [cf. the proof of [GenEll], Lemma 3.5, for more details], that, if, for the sake of simplicity, we ignore the contributions at the archimedean primes, then we obtain relations ht E log(l)  ht E  ht E + log(l) [where we use the notation “” to signify an inequality “≤” up to bounded discrep- ancy, i.e., a relation to the effect that the left-hand side is bounded, independently of the elliptic curve E and the prime number l, by the sum of the right-hand side and some positive real number] and hence, by combining the latter relation “” with the relations of the final displays of (iii), (iv), that ht E  1 l   ht E + log(l) 1 log(l) 1. That is to say, in summary, if, for the sake of i.e., that ht E  l−1 simplicity, we ignore the contributions at the archimedean primes, then the assumption [cf. (iii)] that E admits a GMS implies a bound on the height of E [i.e., ht E ] and hence, in particular, that, if one only considers number fields F of bounded degree over Q, then there are only finitely possibilities for the isomorphism class of E. This is precisely the argument given in [GenEll], Lemma 3.5, which may be regarded as a special case of the argument given in the original proof [due to Faltings] of the invariance of the height [up to bounded discrepancies] under isogenies of abelian varieties. (vi) The argument reviewed in (v) may be understood as consisting of two key points, both of which are closely related to various central aspects of inter-universal Teichmüller theory. The first key point is, of course, (vi-a) the assumption of the existence of a GMS [cf. (iii)], which implies that the passage E  E to the quotient of E by the GMS corresponds to a relation l q E q E between the q-parameters of E and E at each nonarchimedean prime of F where E and E have bad multiplicative reduction i.e., to a relation reminiscent of the Frobenius morphism in positive characteristic. Here, we observe in passing that, in the absence of this crucial assumption of the existence of a GMS, the passage from E to some arbitrary quotient of E by a l at some finite subgroup scheme of rank l would give rise to relations “q E q E LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 79 1/l nonarchimedean primes of bad multiplicative reduction and to relations “q E q E at other nonarchimedean primes of bad multiplicative reduction i.e., a situation in which the argument reviewed in (v) would break down completely! In inter- universal Teichmüller theory, (vi-b) the fundamental role played by theta functions [cf. the discussion of Examples 3.3.2, 3.8.4 below] i.e., in the multiradial reconstruction 2 algorithms of the Θ-pilot “{q j }” means that in addition to a GMS, v it will be necessary to somehow “simulate” [cf. the discussion of Example 3.8.2, (i), below] the existence of a GCG. Here, it is useful to recall that, from the point of view of the classical complex theory of theta functions [cf. also Example 3.3.2 below] +∞  2 q n · U n n=−∞ where q, U C and |q| < 1: · the “U may be understood as the standard multiplicative coor- dinate on the infinite covering “C × E M E” of (ii), hence is only defined once one has a “multiplicative subspace M E[∞]”, i.e., the com- plex analogue of the l-torsion multiplicative subspace “μ l E[l]” of (i), or, alternatively, of the coverings “X K ”, “C K of [Alien], §3.3, (i), (iv), (v) [cf. also the discussion of Example 3.8.2, (i), below]; · the “q” may be understood as the complex q-parameter determined by a generating deck transformation of the finite covering “C × E M E” of (ii), hence is only defined once one has a “generator, up to multiplication by ±1, of E[∞]/M”, i.e., the complex analogue of the l-torsion canonical generator “γ l (E[l]/μ l )(K)”, up to multiplication by ±1, or, alterna- 2 tively, the index “j = 1” in the Θ-pilot “{q j }” of [Alien], §3.3, (vii) [cf. v also the discussion of Example 3.8.2, (i), below; the discussion of Example 3.8.4, (vi), below]. (vii) The second key point of the argument reviewed in (v) (vii-a) consists of the computation of discussed at the beginning of (v) i.e., a computation that essentially amounts to the computation of the logarithmic derivative dlog(U ) = dU U l · dlog(U ) of the isogeny φ, written as “U U l in terms of the standard multi- plicative coordinate “U on G m [cf. (i), (iii)] which, in light of the ampleness of ω M ell [cf. (iv)], implies that “ω M ell | E ω M ell | E [i.e., the “roughly isomorphic” arithmetic line bundles obtained by restrict- ing ω M ell to E, E ] serves, up to a negligible discrepancy, as a common 80 SHINICHI MOCHIZUKI container for the moduli of both E and E , i.e., in light of the existence of the discriminant modular form “Δ M ell [cf. (iv)], as a common l container for both “q E and “q E ”. Here, we observe that this “common container”/ampleness aspect of ω M ell may be understood as corresponding cf. the analogy between étale-like objects in inter-universal Teichmüller theory and tangent bundles/sheaves of differen- tials that was recalled above in the context of (InvHt), (FrDff)! in inter-universal Teichmüller theory, to (vii-b) the theory of the log-link/log-shells and closely related mono-anabelian reconstruction algorithms in a vertical line of the log-theta-lattice that give rise to the log-Kummer-correspondence of inter-universal Teichmüller theory, i.e., which play the fundamental role of furnishing a multiradial container for the Frobenius-like Θ-pilot at the lattice point (0, 0) of the log-theta-lattice [cf. the discussion of §3.6, §3.9, §3.10, §3.11, below!]. Finally, in passing, we note that the discriminant modular form “Δ M ell is also reminiscent of the classical complex theta function of Example 3.3.2 below [i.e., which may also be regarded as a modular form on the upper half-plane], as well as of the discussion of the relationship between the discriminant modular form and scheme-theoretic Hodge-Arakelov theory in the final portion of [HASurI], §1.2 [cf. also [Alien], Example 2.14.3; [Alien], §3.9, (i), (ii), for a discussion of the relationship between scheme-theoretic Hodge-Arakelov theory and inter-universal Teichmüller theory]. (viii) Thus, one may summarize the discussion of the present Example 3.2.1 as follows: At a very rough, introductory/expository level, one may think of inter- universal Teichmüller theory as a sort of generalization of the ap- proach of (InvHt) [cf. (v)] to bounding heights of elliptic curves over number fields to the case of [essentially] arbitrary elliptic curves over number fields [i.e., which are not assumed to admit a GMS!] by · somehow “simulating” a GMS/GCG and · applying the theory of theta functions and mono-anabelian geometry. Example 3.2.2: Coricity, symmetry, and commutativity properties of the log-theta-lattice. In the following discussion, we fix notation as follows: Let k be a finite extension of Q p , for some prime number p; k an algebraic closure of k; q k a nonzero element of the maximal ideal m k of the ring of integers O k of k; N 2 an integer. Write · N for the additive monoid of nonnegative integers; def · G k = Gal(k/k); · O k for the ring of integers of k, with maximal ideal m k O k ; LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 81 · O  O k for the multiplicative monoid of nonzero elements of O k ; k · O × O  for the group of invertible elements of O  ; k k k · O ×  O ×μ , k k ×μ ×  k k μ ×μ , k  k μ for the respective quotients of O × , k k , k by the action of the group μ of torsion elements [i.e., roots of unity] of O × ; k · F n = O ×μ × (q n ) N k def ×μ k for the multiplicative submonoid of k ×μ [equipped with a natural action by G k ] generated by O ×μ and q n , where k we allow n to be an arbitrary positive integer, and, by a slight abuse of ×μ notation, we write “q” for the image of q m k in k ; · Θ k : F N F 1 for the isomorphism of G k -monoids that restricts to the identity isomorphism on O ×μ and maps q N q; k · log k : O ×  k for the p-adic logarithm on O × . k k Thus, G k acts naturally on O ×μ  O × O  O k . Let Π  G k be a topological k k k group equipped with a surjection onto G k , which determines natural actions of Π on O ×μ  O × O  O k . k k k (i) We begin by observing the following properties: (i-a) The isomorphism Θ k : F N F 1 is not compatible with the ring structures in its domain/codomain in the sense that it does not arise from a G k -equivariant ring homomorphism φ : k → k, i.e., there does not exist a G k -equivariant ring homomorphism ×μ ×μ φ : k → k for which the induced map k k restricts either to a map F N F 1 that coincides with Θ k or to a map F 1 F N that coincides with Θ −1 . k (i-b) The map log k : O × k k is not compatible with the ring structures in its domain/codomain in the sense that it does not arise from a G k -equivariant ring homomorphism φ : k → k, i.e., there does not exist a G k -equivariant ring homomorphism φ : k → k that restricts to log k . Indeed, both (i-a) and (i-b) follow immediately from the easily verified elementary fact that any G k -equivariant ring homomorphism φ : k → k induces [by passing to G k -invariants and considering the l-divisibility properties of units/non-units of k, for prime numbers l  = p] an isomorphism of topological fields k k, hence an isomorphism between the value groups of the copies of k in the domain/codomain. [Here, we recall that, since N 2, the assignments q N q [cf. (i-a)] and (O × ) 1 + p 2 log k (1 + p 2 ) (∈ p · O k ) k [cf. (i-b)] 82 SHINICHI MOCHIZUKI yield immediate contradictions to the existence of such an induced isomorphism between value groups.] By contrast, we observe that (i-c) the isomorphism Θ k : F N F 1 is compatible in the sense of equivariance, relative to the natural actions on the domain/codomain of Θ k with an isomorphism G k G k between the copies of G k in the domain/codomain of Θ k ; (i-d) the map log k : O × k is compatible in the sense of equivariance, k relative to the natural actions on the domain/codomain of log k with an isomorphism Π Π between the copies of Π in the domain/codomain of log k . Indeed, (i-c) and (i-d) follow immediately from the various definitions involved. Here, we recall, however, that it is of fundamental importance to observe the fol- lowing: (i-e) Since Θ k and log k are not compatible with the respective ring struc- tures in their domains/codomains [cf. (i-a), (i-b)!], in order to obtain coric structures i.e., structures that are commonly shared, in an in- variant fashion, by these domains/codomains, hence well-defined in a sense that is independent of any specification of a relationship to these domains/codomains it is necessary to regard · the isomorphisms G k G k and Π Π as indeterminate isomorphisms of abstract groups, i.e., not of Galois groups, that is to say, groups equipped with the “Galois-rigidification” constituted by the auxiliary data of some sort of action on a field/ring; · the “identity isomorphism” given by restricting Θ k to O ×μ k O ×μ O ×μ k k as an indeterminate isomorphism of topological monoids equipped with an action by a topological group [i.e., G k ], as well as with the collection of submonoids given by the images H in O ×μ of the intersections O × k [i.e., of O × with the H- k invariants k of G k H k k k of k], where H ranges over the open subgroups cf. the discussion of §3.8 below. These coricity properties will also play an important role in the context of the discussion of (ii-c) below. In the context of (i-e), it is also important to note the following: (i-f) The use of the terminology “identity isomorphism” when referring to any of the isomorphisms G k G k , Π Π, O ×μ O ×μ k k LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 83 discussed in (i-e) can be highly misleading and give rise to unnecessary confusion, for the following reason: Strictly speaking, throughout mathe- matics, this terminology “identity isomorphism” is only well-defined when applied to an isomorphism from a mathematical object to itself [i.e., not to a distinct mathematical object!]. Since, however, the ring structures in the domain/codomain of Θ k or log k must be distinguished [i.e., so long as they are related to one another via Θ k or log k cf. (i-a), (i-b), as well as the discussion of §3.4 below!], the only possible “well-defined sense” in which this terminology “identity isomorphism” may be applied is the sense of referring to some sort of “identity isomorphism” between weaker underlying structures [i.e., such as “sets”, “abstract topological groups”, “abstract topological monoids”, etc. cf. the discussion of §3.8 below] that do indeed coincide, hence may indeed be “identified” with one another [cf. the situations discussed in Example 3.5.2 below]. The observations of (i-e) and (i-f) play a fundamental role in the essential logical structure of inter-universal Teichmüller theory. (ii) Next, we observe the following properties: (ii-a) The isomorphism Θ k : F N F 1 is not symmetric with respect to switching the domain/codomain in the following sense: there do not exist G k -equivariant ring homomor- ×μ phisms φ : k → k, ψ : k → k such that the induced maps φ ×μ : k ×μ ×μ ×μ ×μ k , ψ : k k fit into a diagram ×μ k ×μ F N Θ k −→ F 1 ×μ k ×μ φ k ψ ×μ F 1 Θ k ←− F N k ×μ that is commutative on the portion of the diagram on which the relevant composites are defined. (ii-b) The isomorphism log k : O × k k is not symmetric with respect to switching the domain/codomain in the following sense: there do not exist G k -equivariant ring homomor- phisms φ : k → k, ψ : k → k that fit into a diagram k φ k O × k log k ←− O × k log k −→ k ψ k that is commutative on the portion of the diagram on which the relevant composites are defined. Indeed, (ii-a) follows by tracing the image, at the level of value groups, of the q N in the upper left-hand corner of the diagram, i.e., which maps [cf. the discussion of 84 SHINICHI MOCHIZUKI induced isomorphisms of value groups following (i-a), (i-b)] to q via the composite 2 of the upper horizontal and right-hand vertical arrows, but to q N ( = q N ) via the composite of the left-hand vertical and lower horizontal arrows. In a similar vein, (ii-b) follows by tracing the image, at the level of value groups, of the element 1 u = {exp(p 2 )} p 2 O × def k [where “exp(−)” denotes the well-known formal power series of the exponential function, and the p 1 2 denotes a p 2 -root of the element “{−}”] in the upper left- hand corner of the diagram, i.e., which maps to 0 via the composite of the upper horizontal, right-hand vertical, and lower horizontal arrows, but to u ( = 0) via the left-hand vertical arrow. By contrast, we observe the following: (ii-c) The topological group actions surrounding Θ k [cf. (i-c)] Π  G k  Π   Π  Π  Θ k F N −→ G k yield a diagram G k F 1 G k that is [manifestly!] symmetric with respect to applying the operation of reflection of this last diagram around the ”, where we observe that this symmetry property only holds if the coricity properties discussed in (i-e), (i-f) are applied, i.e., if · the isomorphism of topological groups G k G k is regarded as an indeterminate isomorphism of abstract topological groups and · the arrows “”/“” are regarded as surjections between abstract topological groups, i.e., topological groups that are only well-defined up to indeterminate isomorphism. Indeed, if one does not apply these coricity properties, then one is in effect working with structures [i.e., the various “Π” and “G k in “Π  G k G k  Π”] that depend on Galois-rigidifications that arise from structures specific to the domain or codomain of Θ k , hence do not satisfy the desired symmetry property [cf. the discussion of (i-e), (i-f)!]. The observation (ii-c), together with the observations (i-e) and (i-f), plays a funda- mental role in the essential logical structure of inter-universal Teichmüller theory. (iii) Next, we observe the following properties: (iii-a) Θ k does not commute with log k i.e., at a purely formal level, “Θ k log k  = log k Θ k in the following sense: there do not exist G k -equivariant ring homomorphisms φ : k → k, ψ : k → k such that the LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY μ μ O ×μ k log μ 85 μ induced maps φ μ : k k , ψ μ : k k fit into a diagram F N Θ k −→ F 1 k k φ μ | k O ×μ k log k k ψ μ | k μ F N Θ k −→ F 1 k k μ that is commutative on the portion of the diagram on which the relevant composites are defined. (iii-b) Θ k is not invariant with respect to log k i.e., at a purely formal level, “Θ k log k  = Θ k in the following sense: there do not exist G k - equivariant ring homomorphisms φ : k → k, ψ : k → k such that the μ μ μ μ induced maps φ μ : k k , ψ μ : k k fit into a diagram O ×μ k log F N Θ k −→ F 1 k k k φ μ | k O ×μ k ι μ k ψ μ k μ F N Θ k −→ F 1 k μ μ where ι k : O ×μ → k denotes the natural inclusion that is commu- k tative on the portion of the diagram on which the relevant composites are defined. Indeed, let m be a positive integer such that p m · O k log k (O k × ) p −m · O k , where we write O k × = O k O × . Then (iii-a) follows by tracing the image of the element def k u n = exp(p n·N ) O × def k [where “exp(−)” denotes the well-known formal power series of the exponential function, and n is any positive integer > 2m [which implies that n < n+(n−2m) = 2n 2m n · N 2m] such that p n q N · O × ] in the upper left-hand portion of k the diagram, i.e., which maps [cf. the discussion of induced isomorphisms of value groups following (i-a), (i-b)] to [the image in the value group of k of] some nonzero element p n·N −2m ·O k [cf. the discussion of (i-e), as well as of Example 3.5.1 below] via the composite of the upper horizontal and right-hand vertical arrows, but to [the image in the value group of k of] p n (∈ p n·N −2m · O k ) via the left-hand vertical and lower horizontal arrows. In a similar vein, (iii-b) follows by tracing the image of the same element u n O × in the upper left-hand portion of the diagram, i.e., which k maps [cf. the discussion of induced isomorphisms of value groups following (i-a), (i-b)] to [the image in the value group of k of] u n via the composite of the upper horizontal and right-hand vertical arrows, but to [the image in the value group of k of] p n ( = u n ) via the left-hand vertical and lower horizontal arrows. 86 SHINICHI MOCHIZUKI (iv) Finally, we observe that it is now an essentially formal/routine matter to translate the various elementary properties discussed above in (i), (ii), (iii) into the corresponding coricity/symmetry/commutativity properties of the log-theta- lattice, i.e.: · the incompatibility with the ring structures in the domain/codomain [cf. the discussion at the beginning of the present §3.2; the discussion at the beginning of §3.8 below] of the Θ-link [cf. (i-a)] and log-link [cf. (i-b)]; · the horizontal [cf. (i-c)] and vertical [cf. (i-d)] coricity [cf. (i- e), (i-f)] properties of the étale-like structures that appear in the log- theta-lattice [cf. the discussion at the beginning of the present §3.2; the discussion at the beginning of §3.8 below]; · the non-symmetricity with respect to switching the domain/codomain [cf. the discussion at the beginning of the present §3.2] of the Θ-link [cf. (ii-a)] and log-link [cf. (ii-b)]; · the symmetricity with respect to the Θ-link [cf. the discussion at the beginning of the present §3.2] of the étale-like structures of the log-theta-lattice [cf. (ii-c)]; · the non-commutativity of the log-theta-lattice [cf. (iii-a); the discus- sion at the beginning §3.3 below]; · the non-invariance of the Θ-link with respect to the log-link [cf. (iii- b); the discussion at the beginning §3.3 below]. §3.3. RCS-redundant copies in the domain/codomain of the log-link The Θ-link of inter-universal Teichmüller theory is defined, in the style of classical complex Teichmüller theory [cf. Example 3.3.1 below; [IUTchI], Remark 3.9.3], as a deformation of the ring structure in a ±ell NF-)Hodge theater that depends, in an essential way, on the splitting into unit groups and value groups of the various localizations of the number field involved. On the other hand, the log-link of inter-universal Teichmüller theory [i.e., in essence, the p-adic logarithm at primes of the number field of residue characteristic p] has the effect of juggling/rotating these unit groups and value groups, e.g., by mapping units to non-units [cf., e.g., the discussion of [Alien], Example 2.12.3, (v)]. In particular, there is no natural way to relate the two Θ-links [i.e., the two horizon- tal arrows in the following diagram] that emanate from the domain and codomain of the log-link [i.e., the left-hand vertical arrow in the following diagram] Θ −→ log Θ −→ . .. ?? .. . that is to say, there is no natural candidate for “??” [i.e., such as, for instance, an isomorphism or the log-link between the two bullets “•” on the right-hand side LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 87 of the diagram] that makes the diagram commute. Indeed, it is an easy exercise [cf. Example 3.2.2, (iii), (iv); [Alien], §3.3, (ii); [Rpt2018], §15, (LbΘ), (Lblog), (LbMn)], to show that neither of these candidates for “??” [i.e., an isomorphism or the log-link] yields a commutative diagram. Thus, in summary, any identification of the domain and codomain of the log- link [cf. (RC-log)!] yields a situation in which the local splittings into unit groups and value groups of the resulting identified “•’s” are no longer well-defined. In particular, any such identification of the domain and codomain of the log-link [cf. (RC-log)!] yields a situation in which the Θ-link is not well-defined i.e., a situation in which the apparatus of inter-universal Teichmüller theory completely ceases to function cf. the discussion of the definition of the Θ-link in the latter half of [Alien], §3.3, (ii), as well as the discussion of Example 3.3.3, (i), below. This discussion may be summarized, at a symbolic level, as follows: definition of the Θ-link =⇒ falsity of (RC-log). Next, we observe [cf. the discussion of [IUTchI], Remark 3.9.3, (iii), (iv)] that the non-existence of a solution for “??” in the above diagram [i.e., that makes the diagram commute] amounts, at a structural level, to essentially the same phe- nomenon as the incompatibility of the dilations that appear in classical complex Teichmüller theory with multiplication by non-real roots of unity [cf. Example 3.3.1 below]. Write R, C, respectively, for the topological fields of real and complex numbers. Then as observed in the discussion of the latter half of [Alien], §3.3, (ii) [cf., especially, the discussion surrounding [Alien], Fig. 3.6]: .. . log .. . log Θ Θ Θ Θ Θ Θ .. . log .. . log log log ... −→ Θ −→ −→ . . . log log ... −→ Θ −→ −→ . . . log log ... −→ Θ −→ −→ . . . log log log log .. . .. . .. . .. . Θ −→ log log (InfH) this structural similarity is consistent with the analogy discussed in loc. cit. between · the “infinite H” portion of the log-theta-lattice consisting of the two vertical lines [i.e., of log-links] on either side of a horizontal arrow [i.e., a Θ-link] of the log-theta-lattice and 88 SHINICHI MOCHIZUKI · the elementary theory surrounding the bijection × C × \GL + 2 (R)/C [0, 1) λ 0 0 1 λ−1 λ+1 where λ R ≥1 ; GL + 2 (R) denotes the group of 2 × 2 real matrices of positive determinant; C × denotes the multiplicative group of C, which we regard as a subgroup of GL + 2 (R) via the a b , for a, b R such that (a, b)  = (0, 0); assignment a + ib −b a the domain of the bijection is the set of double cosets. That is to say, · the dilation λ 0 1 0 cf. the dilations that appear in classical complex Teichmüller theory, i.e., as reviewed in Example 3.3.1 below corresponds to the Θ-link portion of an “infinite H” [cf. Example 3.3.2, (iii), below], while · the two copies of the group of toral rotations “C × [e.g., by roots of unity in C × ] on either side of “GL + 2 (R)” which may be thought of as a representation of the holomorphic structures in the domain and codomain of the dilation λ 0 1 0 [cf. the discussion of Example 3.3.1 below] correspond, respectively, to the two vertical lines of log-links in the “infinite H” on either side of the Θ-link [cf. the discussion of Example 3.3.2, (iv), below]. Example 3.3.1: Classical complex Teichmüller theory. Let λ R >1 . Re- call the most fundamental deformation of complex structure in classical complex Teichmüller theory Λ: C C  z = x + iy C def ζ = ξ + = λ · x + iy C where x, y R. Let n 2 be an integer, ω a primitive n-th root of unity. Write ∈) μ n C for the group of n-th roots of unity. Then observe that if n 3, then there does not exist ω  μ n such that Λ(ω · z) = ω  · Λ(z) for all z C. [Indeed, this observation follows immediately from the fact that if n 3, then ω ∈ R.] That is to say, in words, Λ is not compatible with multiplication by μ n unless n = 2 [in which case ω = −1]. This incompatibility with “indeterminacies” arising from multiplication by μ n , for n 3, may be understood as one fundamental reason for the special role played by square differentials [i.e., as opposed to n-th power differentials, for n 3] in classical complex Teichmüller theory [cf. the discussion of [IUTchI], Remark 3.9.3, (iii), (iv)]. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 89 Example 3.3.2: The Jacobi identity for the classical theta function. (i) Write z = x + iy for the standard coordinate on the upper half-plane def H = {z = x + iy C | y > 0}. Recall the theta function on H +∞  def Θ(q) = 1 2 q 2 n . n=−∞ def where we write q = e 2πiz . Restricting to the imaginary axis [i.e., x = 0] yields a function +∞  2 def θ(t) = e −πn t . n=−∞ def where we write t = y. (ii) Next, let us observe that   01 def ι = C × GL + (R) −1 0 maps z −z −1 , hence iy iy −1 , i.e., t t −1 , while, for λ R ≥1 ,   λ 0 C × GL + (R) 01 maps z λ · z, hence iy · y, i.e., t λ · t. (iii) Next, we observe the following: · As t +∞, the terms in the series for θ(t) are rapidly decreasing, and θ(t) +0. In particular, the series for θ(t) is relatively easy to compute. · As t +0, the terms in the series for θ(t) decrease very slowly, and θ(t) +∞. In particular, the series for θ(t) is very difficult to compute. Thus, in summary, the “flow/dilation” λ 0 1 0 along the imaginary axis may be re- garded as a sort of “link”, in the context of the theta function θ(t), between small values [i.e., θ(t) +0 as t +∞] and large values [i.e., θ(t) +∞ as t +0]. That is to say, this flow/dilation along the imaginary axis behaves in a way that is strongly reminiscent of the Θ-link of inter-universal Teichmüller theory [cf. the discussion of (InfH)]. (iv) The Jacobi identity for the theta function θ(t) θ(t) = t 2 · θ(t −1 ) 1 allows one to analyze the behavior of θ(t) as t +0, which is very difficult to compute [cf. (iii)], in terms of the behavior of θ(t) as t +∞, which is relatively easy to compute [cf. (iii)] cf. the discussion of the Jacobi identity in [Pano], §3, 90 SHINICHI MOCHIZUKI §4; [Alien], §4.1, (i). Observe that this identity may be understood as a sort of invariance with respect to ι [cf. (ii)], up to a certain easily computed factor [i.e., 1 t 2 ]. Note that ι “juggles”, or “rotates/permutes”, the two dimensions of R 2 . This aspect of ι is strongly reminiscent of the log-link of inter-universal Teichmüller theory, which “juggles”, or “rotates/permutes”, the two underlying dimensions of the ring structures in a vertical column of the log-theta-lattice [cf., e.g., the discus- sion of [Alien], Example 2.12.3, (v)]. By contrast, we note that the theta function θ(t) does not satisfy any interesting properties of invariance with respect to the dilations λ 0 1 0 . (v) Relative to the analogy with the Θ-link and log-link of inter-universal Teichmüller theory discussed in (iii), (iv), the ι-invariance interpretation of the Jacobi identity discussed in (iv) is strongly reminiscent of the central role played by log-link invariance in the construction of the multiradial representation in inter-universal Teichmüller theory [cf. the discussion surrounding (logORInd), (Di/NDi) in §3.11 below; the discussion of the Jacobi identity in [Pano], §3, §4]. 1 Here, we note that the factor t 2 in the Jacobi identity may be understood as corresponding, relative to the analogy with inter-universal Teichmüller theory, to the indeterminacies (Ind1), (Ind2), (Ind3) acting on the log-shells in the multiradial representation. Indeed, both · the factor t 2 in the Jacobi identity which amounts, in essence, to the well-known interpretation of the theta function θ(t) as a modular form on H and 1 · the log-shells in the multiradial representation cf. the discussion of the relationship between scheme-theoretic Hodge-Arakelov theory and inter-universal Teichmüller theory in [Alien], Example 2.14.3; [Alien], §3.9, (i), (ii) are closely related to the notion of “differentials”. (vi) At a more technical level, the crucial ι-invariance property of (iv), (v) may be understood as a consequence of the Fourier transform invariance of 2 the Gaussian “e − on the real line [cf. the discussion of the Jacobi identity in [Pano], §3, §4]. This Fourier transform invariance in turn may be understood as a consequence of the quadratic form “ 2 in the exponent of the Gaussian 2 “e − ”, which may be thought of as the first Chern class of the ample line bundle whose section determines, via the canonical theta trivialization of the line bundle, the theta function under consideration [cf. the classical theory of complex theta functions as exposed, for instance, in [Mumf1], Chapter I]. When this quadratic form “ 2 is multiplied by a factor t R >0 , application of the Fourier transform gives rise, up to suitable factors [involving, in particular, the Gaussian integral! cf. the discussion of (v), as well as [Alien], §3.8], to the transformation e −t· 2  e −t −1 · 2 that underlies the Jacobi identity. In this context, it is of central importance to observe that the transformation “t  t −1 in the above display [i.e., as opposed to a transformation of the form “t  c · t −1 ”, for some c R >0 ] is indicative of LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 91 and indeed in some sense essentially equivalent to an absolute notion of “1” [i.e., the unique invariant element of the transformation R >0  t t −1 R >0 ] 2 in the copy of the real numbers that appears in the exponent “− 2 of “e − ”. Finally, we observe that the fundamental role played by the quadratic form “ 2 of the above discussion in the proof of the Fourier transform invariance of the Gaussian that underlies the Jacobi identity is strongly reminiscent of the crucial rigidity properties in the theory of the étale theta function [cf. [Alien], §3.4, (iii), (iv); the discussion of the Jacobi identity in [Pano], §3, §4] that underlie the multiradial representation of inter-universal Teichmüller theory: Indeed, these rigidity properties may be understood as consequences of the theta group symmetries, which also arise, in essence, from the étale-theoretic version of the quadratic form “ 2 of the above discussion [cf. the discussion at the end of [Alien], §3.4, (iv)]. (vii) Before continuing our discussion, we pause briefly to make the following elementary observation: Let V be a 1-dimensional R-vector space. Then a [topological] ring/field structure on V may be understood as a multiplication map V R V V given by an isomorphism of R-vector spaces. By tensoring with the dual vector space to V , one verifies immediately that such a multiplication map may be understood as an isomorphism of R-vector spaces V R, i.e., as the choice of a nonzero element in V given by the image of 1 R. In particular, it follows immediately from this elementary observation that the absolute notion of “1 R” discussed in (vi) may be interpreted as a [topological] ring/field structure on the copy of the real numbers that appears in the exponent “(−)” of the Gaussian “e (−) ”. This interpretation is strongly reminiscent of the central importance, in inter-universal Teichmüller theory, of working with the first power of [the reciprocal of the l-th root of] the theta function [cf., e.g., the discussion of [Alien], §3.4, (iii)], which makes it possible to consider the truncated “mod N Kummer theory of the theta function: indeed, this truncatibility of the Kummer theory of the theta function is closely related to the [topological] ring/field structure of the local fields that appear in the context of the log-link of inter-universal Teichmüller theory [cf. the analogy between the log-link and “ι” discussed in (iv); the discussion of Example 3.8.4 below; the discussion of the final portion of [Alien], §3.6, (ii)]. (viii) Another important technical aspect of the Fourier transform discussed in (vi) is the factor “e ixy [where x, y R] that appears in this Fourier transform. Indeed, · this “exponentiation of a complex unit S 1 C × to a power given by some indeterminate real number i.e., the real number that corresponds to the variable of integration in the Fourier ransform is strongly reminiscent of · the (Ind2) indeterminacy action on the local units “O ×μ in inter- universal Teichmüller theory i.e., which amounts, in essence, to expo- nentiation of these local units to a power given by some indeterminate  × element Z 92 SHINICHI MOCHIZUKI [cf. the discussion of the Jacobi identity in [Pano], §3, §4]. Moreover, in this context, it is of interest to note that the integration that occurs in the Fourier ˙ transform may be understood as corresponding to the logical OR/XOR “∨/ ∨” aspect of the indeterminacies that occur in inter-universal Teichmüller theory ˙ with addition [where we recall that cf. the correspondence of logical XOR ∨” “integration” may be understood as a sort of “topological addition” operation], as ˙ discussed in Example 2.4.6, (iii), as well as in (∧( ∨)-Chn) in §3.10 below. Example 3.3.3: Theta functions and multiplicative structures. (i) One fundamental reason for the central role played by theta functions in the essential logical structure of inter-universal Teichmüller theory lies in the fact that (ThMlt) the main properties of interest of the theta functions that appear in inter-universal Teichmüller theory most notably, (ZrPl) the well-known description of the zeroes/poles of theta func- tions at the cusps [cf. [EtTh], Proposition 1.4, (i); [IUTchIII], Remark 2.3.3, (vi), (vii); [Alien], §3.4, (iii)], which yields by applying the well-known intersection theory of divisors supported on the special fiber of the universal topological covering of the Tate curve [cf. the discussion preceding [EtTh], Proposition 1.1, of divisors supported on the special fiber] a “divisor-theoretic” characterization of these theta functions [up to translation by a deck transformation of the universal topological covering]; (SymTh) the well-known symmetries of theta functions [cf. [EtTh], Proposition 1.4, (ii)]; (GalEv) the Kummer-compatible Galois evaluation properties of theta functions [cf. [IUTchII], Remark 1.12.4; the discussion at the beginning of [Alien], §3.6; [EtTh], Remark 1.10.4, (i)], which give rise to the canonical splittings of theta monoids, as well as to the construction of the Gaussian monoids [cf. [IUTchII], Corol- laries 2.5, 2.6, 3.5, 3.6; [Alien], §3.4, (iii); [Alien], §3.6, (ii)] may be expressed entirely in terms of the multiplicative structures of the various rings that appear, i.e., without invoking the additive struc- tures of these rings. This expressibility purely in terms of multiplicative structures plays an essential role in establishing the · multiradial unit group/value group splittings/decouplings [cf. [IUTchII], Remark 1.12.2, (vi); [Alien], §3.4, (iii)] and · non-interference properties [cf. [Alien], §3.7, (i)] that underlie the definition of the Θ-link [cf. the discussion of the present §3.3 pre- ceding (InfH)] and log-Kummer-correspondence [cf. [IUTchII], Remark 1.12.2, (iv); [IUTchIII], Remark 1.2.3]. (ii) In the context of (ZrPl), it is important to recall the central importance of the fact that LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 93 (ZrPlOrd) the signed order [i.e., “+” for zeroes, “−” for poles] of the theta function at each of the cusps is precisely one [cf. [IUTchIII], Remark 2.3.3, (vi), (vii); [Alien], §3.4, (iii)]. This property (ZrPlOrd) of the theta functions that appear in inter-universal Te- ichmüller theory is closely related to the properties discussed in Example 3.3.2, (vi), (vii), in the case of complex theta functions. In inter-universal Teichmüller theory, this property (ZrPlOrd) ensures that the mono-theta-theoretic cyclo- tomic rigidity algorithms that arise from the theory of theta functions are · compatible with the topology of the tempered fundamental group and, moreover, are · not subject to {±1}-indeterminacies properties that are not satisfied by the cyclotomic rigidity isomorphisms that arise from the theory of algebraic rational functions [cf. [IUTchIII], Remark 2.3.3, (vi), (vii); [IUTchIII], Remark 3.11.4; [Alien], §3.4, (iii), as well as the discussion of Examples 3.3.2, (vi), (vii); 3.8.4, (iv), (v), (vi), below, of the present paper]. Here, it is of interest to recall that (MltAdd) the anabelian reconstruction algorithms of [AbsTopIII], Theorem 1.9, im- ply that the additive structure of the function field of an algebraic curve may in fact be reconstructed from the multiplicative structure of the func- tion field of algebraic rational functions on the curve, equipped with the [multiplicative!] valuation maps and [multiplicative!] evaluation maps at the closed points of the curve [cf. “Uchida’s Lemma”, i.e., [AbsTopIII], Proposition 1.3]. That is to say, (MltAdd) means that once one allows oneself to work with algebraic rational functions i.e., as opposed to theta functions the issue emphasized in (i) of expressibility purely in terms of multiplicative structures in some sense ceases to be well-defined/non-vacuous. By contrast, if one restricts oneself, as is indeed the case in the discussion of (i), to considering theta functions as is necessary, in order to apply the essential property (ZrPlOrd) discussed above! then (MltAdd) is no longer applicable, so there are no longer any obstructions to the “well-definedness/non-vacuousness” of the notion of “expressibility purely in terms of multiplicative structures”. Finally, we recall that, in any vertical line of log-links in the log-theta-lattice, · the discrepancy between the [holomorphic] Frobenius-like copies of objects on either side of a log-link [cf. (RC-log)], as well as · the discrepancy between [holomorphic] Frobenius-like copies of objects and [holomorphic] étale-like copies of objects [cf. (RC-FrÉt)], may be understood as the extent to which the diagram of arrows that constitutes the log-Kummer-correspondence associated to this vertical line of log-links fails to commute. This failure to commute may be estimated by means of the indeterminacy (Ind3), i.e., by interpreting this failure to commute as a sort of “upper semi- commutativity”. This indeterminacy (Ind3) is highly nontrivial and, in particu- lar, gives rise to the inequality that appears in the final computation of log-volumes 94 SHINICHI MOCHIZUKI in inter-universal Teichmüller theory [cf. [IUTchIII], Corollary 3.12]. In this con- text, it is important to recall that the theory surrounding this indeterminacy (Ind3) depends, in an essential way, on the absolute anabelian geometry of [AbsTopIII], §1, i.e., which allows one to reconstruct a hyperbolic curve X over a number field or mixed characteristic local field from the abstract profinite group determined by the étale fundamental group π 1 (X) of the curve. That is to say, in summary, this absolute anabelian geometry allows one to show that the discrepancies between the various [holomorphic] Frobenius-like and [holomorphic] étale-like copies of objects in a vertical line of log-links [cf. (RC-log), (RC-FrÉt)] in the log-theta-lattice are “bounded by” the [rel- atively mild] indeterminacy (Ind3). On the other hand, this absolute anabelian geometry most certainly does not imply that these discrepancies are trivial/non-existent, i.e., as asserted in (RC-log), (RC- FrÉt) cf. the discussion of the falsity of (RC-log), (RC-FrÉt) in §3.2 and the present §3.3. §3.4. RCS-redundant copies in the domain/codomain of the Θ-link The Θ-link of inter-universal Teichmüller theory Θ −→ is defined as a gluing between the ±ell NF-)Hodge theater “•” in the domain of the arrow and the ±ell NF-)Hodge theater “•” in the codomain of the arrow along F ×μ -prime-strips “∗” that arise from the Θ-pilot object “Θ-plt” in the domain and the q-pilot object “q-plt” in the codomain. Here, it is important to note that this gluing is obtained by regarding these F ×μ -prime-strips “∗” as being known only up to isomorphism. This point of view, i.e., of regarding these F ×μ -prime-strips “∗” as being known only up to isomorphism, is implemented formally by taking the gluing to be the full poly-isomorphism i.e., the set of all isomorphisms between the F ×μ -prime-strips arising from the domain and codomain of the Θ-link. Here, we recall that · q-plt essentially amounts to the arithmetic line bundle determined by [the ideal generated by] some 2l-th root q of the q-parameter at the valuations v v V bad , while · Θ-plt essentially amounts to the collection of arithmetic line bundles de- 2 termined by [the ideals generated by] the collection {q j }, as j ranges over v def the integers 1, . . . , l  = l−1 2 [where l is the prime number that appears in the initial Θ-data under consideration], and v ranges over the valuations V bad . Also, we recall that each ±ell NF-)Hodge theater “•” gives rise to an associated model “Ring” of the ring/scheme theory surrounding the elliptic curve under con- sideration. In the following discussion, we shall write · for the “•” in the domain of the Θ-link, LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 95 · for the “•” in the codomain of the Θ-link, ·  for an arbitrary element of the set consisting of “†”, “‡”, and the “empty symbol” [i.e., no symbol at all], ·  Θ-plt  Ring for the Θ-pilot arising from the collection “{  q j }” that 2 v appears in the model of ring/scheme theory associated to  •, and ·  q-plt  Ring for the q-pilot arising from the  q that appears in the model of ring/scheme theory associated to  •. v Finally, we recall that since, for j  = 1, the valuation [at each valuation v V bad ] 2 of q j differs from that of q , the arithmetic degrees of the line bundles constituted v by q-plt and Θ-plt differ. v Thus, at a more formal level, the above description of the gluing that consti- tutes the Θ-link may be summarized as follows: Ring  Θ-plt ←: :→ q-plt Ring Ring  q-plt  = Θ-plt Ring [where “←:” and “:→” denote the assignments that consitute the gluing discussed above]. In this context, we note the following fundamental observation, which un- derlies the entire logical structure of inter-universal Teichmüller theory [cf. the discussion of [IUTchIII], Remark 3.12.2, (c itw ), (f itw ); [Alien], §3.11, (iv)]: (AOΘ1) the following condition holds:     :→ Θ-plt Ring :→ q-plt Ring . By contrast, if one simply deletes the distinct labels “†”, “‡” [cf. (RC-Θ)!], then (AOΘ2) the following condition holds:     :→ Θ-plt Ring :→ q-plt Ring . Of course, (AOΘ3) the essential mathematical content discussed in this condition (AOΘ2) may be formally described as a condition involving the AND relator “∧”:     q-plt {q-plt, Θ-plt} Θ-plt {q-plt, Θ-plt} . On the other hand, precisely as a consequence of the fact [discussed above] that Ring  q-plt  = Θ-plt Ring, 96 SHINICHI MOCHIZUKI (AOΘ4) the following condition does not hold:     :→ Θ-plt Ring :→ q-plt Ring . That is to say, the operation of identifying •, [hence also Ring, Ring] e.g., on the grounds of “redundancy” [i.e., as asserted in (RC-Θ)!] by deleting the distinct labels “†”, “‡” has the effect of passing from a situation in which the AND relator “∧” holds [cf. (AOΘ1)] to a situation in which the OR relator “∨” holds [cf. (AOΘ2), (AOΘ3)], but the AND relator “∧” does not hold [cf. (AOΘ4)]! In particular, relative to the correspondences •, Ring ←→ I; Θ-plt ←→ ←→ β; γ J ; q-plt •, Ring ←→ ←→ I α [cf. the correspondences (StR1) (StR6) discussed in §3.2; the correspondences discussed in Example 2.4.5, (ii); the discussion of [Alien], §3.11, (iv)], one obtains very precise structural resemblances (AOΘ1) (AOΘ2) (AOΘ3) (AOΘ4) ←→ ←→ ←→ ←→ (AOL1), (AOL2), (AOL3), (AOL4) with the situation discussed in Example 2.4.1, (i), (ii). Thus, in summary, the falsity of (RC-Θ) may be understood as a consequence of the falsity [cf. (AOΘ4)] of the crucial AND relator “∧” in the absence of distinct labels, in stark contrast to the truth [cf. (AOΘ1)] of the crucial AND relator “∧” as an essentially tautological consequence of the use of the distinct labels “†”, “‡”. In the context of the central role played in the logical structure of inter-universal Teichmüller theory by the validity of (AOΘ1), it is important to note [cf. the property discussed in (AOΘ4)!] that (NoRng) there does not exist an isomorphism of ring structures Ring Ring that induces, on value groups of corresponding local rings, the desired 2 assignment { q j } q [i.e., that appears in the Θ-link]. v v On the other hand, if, instead of considering the full ring structures of Ring, Ring, one considers [cf. the discussion of [Rpt2018], §6] · certain suitable subquotients i.e., in the notation of [Alien], §3.3, (vii), (a q ), (a Θ ), “O × of the underlying multiplicative monoids k LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 97 of corresponding local fields, as well as · the absolute Galois groups i.e., in the notation of [Alien], §3.3, (vii), (a q ), (a Θ ), “G k associated to corresponding local rings, regarded as abstract topological groups [that is to say, not as Galois groups, or equivalently/alternatively, as groups of field automorphisms! cf. the discussion of §3.8 below], then one obtains structures i.e., the structures that constitute the F ×μ -prime- strips that appear in the Θ-link that are simultaneously associated [as “un- derlying structures”] to both Ring and Ring via isomorphisms [i.e., of certain suitable multiplicative monoids equipped with actions by certain suitable abstract topological groups] that restrict, on the subquotient monoids that correspond to the 2 respective value groups, to the desired assignment { q j } q . It is this crucial v v simultaneity that yields, as a tautological consequence, the validity of the AND relator “∧” in (AOΘ1). Working, as in the discussion above, with multiplicative monoids equipped with actions by abstract topological groups, necessarily gives rise to certain inde- terminacies, called (Ind1), (Ind2), that play an important role in inter-universal Teichmüller theory. Certain aspects of these indeterminacies (Ind1), (Ind2) will be discussed in more detail in §3.5 below. In this context, we recall that one central assertion of the RCS [cf. the discussion of Example 3.2.2; the discussion of (SSInd), (SSId) in [Rpt2018], §7, §10] is to the effect that (NeuRng) these indeterminacies (Ind1), (Ind2) may be eliminated, without affecting the essential logical structure of inter-universal Teichmüller theory, by tak- ing the multiplicative monoids and abstract topological groups that appear in the F ×μ -prime-strips of the above discussion to be equipped with rigidifications by regarding them as arising from some fixed “neutral” ring structure  Ring. On the other hand, as discussed in (NoRng) above, there does not exist any ring structure that is compatible [i.e., in the sense discussed in (NoRng)], with the desired 2 assignment { q j } q . That is to say, in summary, v v (NeuORInd) working with such a fixed “neutral” ring structure  Ring as in (NeuRng) means either that (NeuORInd1) there is no relationship between “∗” and  Ring, or that (NeuORInd2) the relationship between “∗” and  Ring is always necessarily subject to an indeterminacy [cf. (AOΘ2), (AOΘ3)!]         :→ Θ-plt Ring :→ q-plt Ring [cf. the situation discussed in Example 3.2.2; the situation discussed in [Rpt2018], §10, (SSId)]. Here, we observe that whichever of these “options”/“indeterminacies” that appear in (NeuORInd) [i.e., (NeuORInd1), (NeuORInd2)] one chooses to adopt, one is forced to contend with an indeterminacy that is, in some sense, much more drastic 98 SHINICHI MOCHIZUKI than the relatively mild indeterminacies (Ind1), (Ind2) whose elimination formed the original motivation for the introduction of  Ring! Finally, we observe that this much more drastic indeterminacy (NeuORInd) means [cf. the discussion of Example 2.4.4!] that throughout any argument, one must always take the position that the only possible relationship between “∗” and  Θ-plt,  q-plt is one in which (PltRel) “∗” maps either to  Θ-plt or i.e., “∨”! to  q-plt, but not both! Since Ring may be thought of as a ring structure in which “∗” tautologically maps to Θ-plt, while Ring may be thought of as a ring structure in which “∗” tauto- logically maps to q-plt, one may rephrase the above observation as the observation that one must always take the position that the only possible relationship between  Ring, on the one hand, and Ring, Ring, on the other, is one in which (RngRel) the ring structure  Ring is identified either with the ring structure Ring or i.e., “∨”! with the ring structure Ring, but not both! At this point, let us recall [cf., e.g., the discussion of §3.5, §3.11, below; [Rpt2018], §9, (GIUT), (ΘCR)] that inter-universal Teichmüller theory requires, in an essential way, the use of the log-links, hence, in particular, [in order to define the power series of the various p-adic logarithm functions that constitute these log-links!] the ring structures Ring, Ring on both sides i.e., “∧”! of the Θ-link [cf. the discussion surrounding (InfH) of the two vertical lines of log-links in the “infinite H” on either side of the Θ-link]. In particular, we conclude formally that it is impossible to implement the arguments of inter-universal Teichmüller theory once this sort of much more drastic indeterminacy (NeuORInd) has been imposed. §3.5. Gluings, indeterminacies, and pilot discrepancy As discussed in §3.4, the Θ-link involves a gluing { q j } q 2 v v that identifies q [i.e., 2l-th roots of the q-parameters at primes of multiplicative v reduction of the [copy belonging to Ring of the] elliptic curve under consideration] 2 with elements, i.e., the q j ’s, which, when j  = 1, have different valuations from the valuation of q . v v On the other hand, in inter-universal Teichmüller theory, by applying the mul- tiradial representation of [IUTchIII], Theorem 3.11, which involves various in- determinacies (Ind1), (Ind2), (Ind3), and then forming [cf. [IUTchIII], Corollary 3.12, and its proof] the holomorphic hull of the union of possible images of the Θ-pilot in this multiradial representation, LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 99 (ΘGl) one may treat both sides of the Θ-link gluing of the above display as belonging to a single ring theory without disturbing [cf. the crucial AND relator “∧” property discussed in §3.4!] the gluing. Alternative ways to understand the essential mathematical content of (ΘGl) include the following: (NonInf) One may think of (ΘGl) as a statement concerning the mutually non- interference or simultaneous executatibility of the Kummer theo- ries surrounding the q-pilot and Θ-pilot relative to the gluing of abstract F ×μ -prime-strips constituted by the Θ-link, i.e., when the Kummer theory surrounding the q-pilot is held fixed, and one allows the Kummer theory surrounding the Θ-pilot to be subject to various indeterminacies. (Cohab) One may think of (ΘGl) as a statement concerning the “cohabitation”, or “coexistence”, of the q-pilot and Θ-pilot relative to the gluing of abstract F ×μ -prime-strips constituted by the Θ-link within the common container obtained by applying the multiradial representation of the Θ-pilot, forming the holomorphic hull [relative to the holomorphic structure [i.e., ±ell NF-)Hodge theater] that gave rise to the q-pilot under consideration], and finally taking log-volumes. In this context, it is important to recall that this sort of phenomenon i.e., of computations of global degrees/heights of elliptic curves in sit- uations where a certain “confusion”, up to suitable indeterminacies, is allowed between q-parameters of the elliptic curves and certain large positive powers of these q-parameters [i.e., as in (ΘGl)] may be seen in various classical examples such as · the proof by Faltings of the invariance of heights of abelian varieties under isogeny [cf. the discussion of [Alien], §2.3, §2.4, as well as the discussion of Example 3.2.1 in the present paper], · the classical proof in characteristic zero of the geometric version of the Szpiro inequality via the Kodaira-Spencer morphism, phrased in terms of the theory of crystals [cf. the discussion of [Alien], §3.1, (v)], and · Bogomolov’s proof over the complex numbers of the geometric version of the Szpiro inequality [cf. the discussion of [Alien], §3.10, (vi)] cf. also the discussion of [Rpt2018], §16. Moreover, in the case of crystals, we observe that, relative to the notation introduced in Example 2.4.5, (v), (vi), we have correspondences as follows: (CrAND) The logical AND “∧” that appears in the multiradial representation of the Θ-pilot in IUT (= AND-IUT) may be understood as being analogous to the fact that crystals, i.e., “∧-crystals”, may be thought of as objects [on infinitesimal neighborhoods of the diagonal inside products of two copies of the scheme under consideration] that may be simultaneously interpreted, up to isomorphism, as pull-backs via one projection morphism and [cf. “∧”!] as pull-backs via the other projection morphism [cf. the ˙ discussion of (∧( ∨)-Chn1) in §3.10 below; the discussion of [Alien], §3.11, 100 SHINICHI MOCHIZUKI (iv), (2 and ), concerning the interpretation of the discussion of crystals in [Alien], §3.1, (v), (3 KS ), in terms of the logical relator “∧”]. (CrOR) Thus, from the point of view of the analogy discussed in (CrAND), the logical OR “∨” that appears throughout OR-IUT may be understood as corresponding to working with “∨-crystals” [i.e., as opposed to crystals (= ∧-crystals)], that is to say, with objects [on infinitesimal neighbor- hoods of the diagonal inside products of two copies of the scheme under consideration] that may be interpreted, up to isomorphism, as pull-backs via one projection morphism or [cf. “∨”!] as pull-backs via the other pro- jection morphism. Here, we observe that this defining “∨” condition of an ∨-crystal is essentially vacuous since one may obtain ∨-crystals from ar- bitrary objects on the scheme under consideration simply by pulling back such an object to the infinitesimal neighborhood of the diagonal under consideration via one of the two projection morphisms. (CrRCS) In a similar vein, from the point of view of the analogies discussed in (CrAND) and (CrOR), RCS-IUT may be understood as corresponding to the modified version of the usual theory of crystals obtained by re- placing the infinitesimal neighborhoods of the diagonal inside products of two copies of the scheme under consideration [i.e., that appear in the usual theory of crystals!] by the diagonal itself. Such a replacement clearly renders the usual theory of crystals trivial/meaningless, in a fashion that is essentially very similar to the triviality of ∨-crystals dis- cussed in (CrOR). Finally, we observe that this similarity between the modified versions of the usual theory of crystals discussed in (CrOR) and the present (CrRCS) is entirely analogous to the equivalence OR-IUT ⇐⇒ RCS-IUT observed in Example 2.4.5, (v), (XOR/RCS). Unfortunately, however, the situation summarized above in (ΘGl) has resulted in certain frequently voiced misunderstandings by some mathematicians. One such frequently voiced misunderstanding is to the effect that (CnfInd1+2) the situation summarized in (ΘGl) may be explained as a consequence of a “confusion” between q-parameters and large positive powers of these q-parameters that results from the indeterminacies (Ind1), (Ind2). In fact, however, as discussed in Example 3.5.1, (iii), below, at least in the case of q-parameters of sufficiently small valuation [i.e., sufficiently large positive order, in the sense of loc. cit.], such a “con- fusion” [i.e., between q-parameters and large positive powers of these q-parameters] can never occur as a consequence of (Ind1), (Ind2), i.e., both of which amount to automorphisms of the [underlying topological module of the] log-shells involved [cf. also the discussion of (ΘInd) in [Rpt2018], §11]. In this context, we note that this misunderstanding (CnfInd1+2) appears to be caused in many cases, at least in part, by a more general misunderstanding concerning the operation of passage to underlying structures [cf. Example 3.5.2 below]. A more detailed discussion of the operation of passage to underlying structures may be found in §3.9 below. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 101 As discussed in [Rpt2018], §11, the “confusion” summarized in (ΘGl) occurs in inter-universal Teichmüller theory as a consequence not only of the local in- determinacies (Ind1), (Ind2), (Ind3), but also of the constraints imposed by the global realified Frobenioid portions of the F ×μ -prime-strips that appear in the Θ-link. In this context, it is of particular importance to observe that (CnfInd3) the indeterminacy (Ind3), which constrains one to restrict one’s atten- tion to upper bounds [i.e., but not lower bounds!] on the log-volume that is the subject of the computation of [IUTchIII], Corollary 3.12, al- ready by itself i.e., without considering (Ind1), (Ind2), or global reali- fied Frobenioids! [cf. the discussion of (Ind3>1+2) in §3.11 below] is sufficient to account for the possibility of a “confusion” of the sort summarized in (ΘGl) [i.e., between q-parameters and large positive powers of these q-parameters]. Indeed, the indeterminacy (Ind3) is defined in precisely such a way as to identify the ideals generated by arbitrary positive powers of the q-parameters. Example 3.5.1: Bounded nature of log-shell automorphism indetermina- cies. Write Z p for the ring of p-adic integers, for some prime number p; Q p for the field of fractions of Z p . (i) Let M be a finitely generated free Z p -module, which, in the following dis- def cussion, we shall think of as being embedded in M Q p = M Z p Q p ; α : M M an automorphism of the Z p -module M . For n Z, write def U (M, n) = {x M Q p | x p n · M, x ∈ p n+1 · M } M Q p . Then observe that α induces a bijection U(M, n) U(M, n) for every n Z. (ii) In the notation of (i), suppose, for simplicity, that p is odd. Let k be a finite field extension of Q p . Write O k k for the ring of integers of k; O k × O k for the group of units of O k ; m k O k for the maximal ideal of k; I k k for the log-shell associated to k [cf., e.g., the discussion of [IUTchIII], Remark 1.2.2, (i)], i.e., the result of multiplying by p −1 the image log p (O k × ) of O k × by the p-adic logarithm log p (−). Thus, O k I k p −c · O k for some nonnegative integer of c that depends only on the isomorphism class of the field k [cf. [IUTchIV], Proposition 1.2, (i)]. In particular, there exists a positive integer s that depends only on the isomorphism class of the field k such that for any automorphism φ : I k I k 102 SHINICHI MOCHIZUKI of the Z p -module I k and any n Z, it holds that φ(U (O k , n)) s  U(O k , n + i) i=−s [where i ranges over the integers between −s and s]. (iii) In the situation of (ii), we define the order of a nonzero element x k to be . One thus concludes from the final the unique n Z such that x m nk , x ∈ m n+1 k portion of the discussion of (ii) that there exists a positive integer t that depends only on the isomorphism class of the field k such that for any automorphism φ : I k I k of the Z p -module I k and any nonzero element q O k [i.e., such as the q-parameter of a Tate curve over k!], the absolute value of the difference between the orders of q and φ(q) is t, i.e., in words, automorphisms of the Z p -module I k only give rise to bounded discrep- ancies in the orders of nonzero elements of O k . Example 3.5.2: Examples of gluings. Distinct auxiliary structures on some common [i.e., “∧”!] underlying structure may be thought of as gluings of the dis- tinct auxiliary structures along the common underlying structure. Here, we observe that, in general, distinct auxiliary structures on a common underlying structure are not necessarily mapped to one another by some automorphism of the common un- derlying structure. Concrete examples of these generalities may be found in quite substantial abundance throughout arithmetic geometry and include, in particular, the examples (i), (ii), (iii), (iv), (v) given below, as well as the elementary Examples 2.3.2, 2.4.1, 2.4.2, 2.4.3, 2.4.7, 2.4.8, 3.3.1 discussed in §2.3, §2.4, §3.3 [cf. also the discussion of [Rpt2018], §11]. In passing, we observe that these examples may also be understood as interesting examples of the sort of gluing/logical AND “∧” relation that appears in the Θ-/log- links of inter-universal Teichmüller theory, i.e., examples of situations that are qualitatively similar to the Θ-/log-links of inter-universal Teichmüller theory in the sense that they involve distinct auxiliary structures that are glued together along some common auxiliary structure [cf. the discussion of (StR1) (StR6) in §3.2; the discussion of §3.4; the portion of the present §3.5 preceding Example 3.5.1]. (i) The group structures of the finite abelian groups Z/2Z×Z/2Z and Z/4Z are not mapped to one another by any isomorphism of sets, despite the fact that the underlying sets of these two groups are indeed isomorphic to one another. This example is also of interest in light of the discussion of truncated Witt vectors in Example 2.4.6, (iii). (ii) The scheme structures of non-isomorphic algebraic curves over a com- mon algebraically closed field are not mapped to one another by any isomorphism LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 103 of topological spaces, despite the fact that the underlying topological spaces of al- gebraic curves over a common algebraically closed field are indeed isomorphic to one another. (iii) The holomorphic structures of non-isomorphic compact Riemann sur- faces R 1 , R 2 with homeomorphic underlying topological spaces are not mapped to one another by any homeomorphism, i.e., by any isomorphism of topological spaces, despite the fact that the underlying topological spaces of such Riemann surfaces R 1 , R 2 are indeed isomorphic to one another. [This example is in fact essentially similar to the situation discussed in Example 3.3.1, except that in the situation of Example 3.3.1, the “two” Riemann surfaces involved are both isomorphic to the complex plane, hence, in particular, isomorphic to one another.] In this context, it is of interest to observe that (iii-a) if one defines the genus of such a compact Riemann surface R i , for i {1, 2}, as the complex dimension of the space of global holomorphic differentials on the Riemann surface, then it is by no means clear that R 1 and R 2 have the same genus, i.e., since this definition of the genus depends, in an essential way, on the holomorphic structure of the Rie- mann surface. On the other hand, once one verifies that (iii-b) this “holomorphic definition” of the genus coincides with the genus de- fined in terms of the singular homology group of the underlying topo- logical space, it follows immediately that R 1 and R 2 do indeed have the same genus. This contrast between the “holomorphic” and “topological” definitions of the genus is of interest in the context of inter-universal Teichmüller theory since it illustrates (iii-c) the very substantive significance of formulating the definitions of objects or constructions [i.e., in the present discussion, the “genus”] in terms of structures that are coric for the “gluing/link” [i.e., in the present dis- cussion, a comparison of distinct holomorphic structures on homeomor- phic topological spaces] under consideration, that is to say, in terms of structures that are commonly shared in an invariant fashion by, hence satisfy a logical AND “∧” relation relative to, the two objects [i.e., in the present discussion, R 1 and R 2 ] that are related to one another by the link under consideration. We refer to the discussion of §3.8 below for a more detailed treatment of the im- portance of coric structures in inter-universal Teichmüller theory. (iv) The field structures of non-isomorphic mixed-characteristic local fields [which, by local class field theory, may be regarded as [the formal union with “{0}” of] some suitable subquotient of their respective absolute Galois groups] are not, in general, mapped to one another by any isomorphism of profinite groups between the respective absolute Galois groups [cf., e.g., [Ymgt], §2, Theorem, for an example of this phenomenon]. (v) In the notation of Example 3.5.1, (i), let X be a proper smooth curve of def genus 2 over F p = Z p /pZ p . Thus, X may be thought of as an “underlying 104 SHINICHI MOCHIZUKI structure” associated to any lifting of X to Z p , i.e., any flat Z p -scheme Y equipped an isomorphism of F p -schemes Y × Z p F p X. Then observe that non-isomorphic liftings of X to Z p are not, in general, mapped to one another by any automorphism of the F p -scheme X. [Indeed, this is particularly easy to see if one chooses X such that X does not admit any nontrivial automorphisms.] In passing, we note that this example may be regarded as a sort of p-adic analogue of the example of (iii). Finally, we make the important observation that crystals on X [relative to Z p ] are objects that are coric/common to arbitrary liftings of X to Z p . This observation is of particular importance in light of the strong structural resemblances between inter-universal Teichmüller theory and the theory of crystals [cf. [Alien], §3.1, (v); the discussion of (CrAND) in the present §3.5; the discussion of §3.10 below]. §3.6. Chains of logical AND relations From the point of view of the simple qualitative model of inter-universal Te- ichmüller theory given in Example 2.4.5, the discussion of §3.4 concerns the AND relator “∧” in the “Θ-link” portion of Example 2.4.5, (ii). On the other hand, strictly speaking, this portion of inter-universal Teichmüller theory only concerns the initial definition of the Θ-link. That is to say, the bulk of the theory developed in [IUTchI-III] concerns, from the point of view of the simple qualitative model of inter-universal Teichmüller theory given in Example 2.4.5, (ii), the preservation of the AND relator “∧” as one passes from · the “Θ-link” portion of Example 2.4.5, (ii), to · the “multiradial representation” portion of Example 2.4.5, (ii). By contrast, the passage from the “multiradial representation” portion of Example 2.4.5, (ii), to the “final numerical estimate” portion of Example 2.4.5, (ii) i.e., which corresponds to the passage from [IUTchIII], Theorem 3.11, to [IUTchIII], Corollary 3.12 is [cf. the discussion of the final portion of Example 2.4.5, (ii)!] relatively straightforward [cf. the discussion of §3.10, §3.11, below]. At this point, it is perhaps of interest to consider “typical symptoms” of mathematicians who are operating under fundamental misunderstandings con- cerning the essential logical structure of inter-universal Teichmüller theory. Such typical symptoms, which are in fact closely related to one another, include the following: (Syp1) a sense of unjustified and acutely harsh abruptness in the pas- sage from [IUTchIII], Theorem 3.11, to [IUTchIII], Corollary 3.12 [cf. the discussion of the final portions of Example 2.4.5, (ii), (iii)!]; (Syp2) a desire to see the “proof” of some sort of commutative diagram or “compatibility property” to the effect that taking log-volumes of pilot objects in the domain and codomain of the Θ-link yields the same real number [a property which, in fact, can never be proved since it is false! cf. the discussion of §3.5]; (Syp3) a desire to see the inequality of the final numerical estimate obtained as the result of concatenating some chain of intermediate inequali- ties, i.e., as is often done in proofs in real/complex/functional analysis or analytic number theory. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 105 Here, it should be noted that (Syp2) and (Syp3) often occur as approaches to mitigating the “harsh abruptness” of (Syp1). With regard to (Syp3), it should be emphasized that it is entirely unrealistic to attempt to obtain the inequality of the final numerical estimate as the result of concatenating some chain of intermediate inequalities since this is simply not the way in which the logical structure of inter-universal Teichmüller theory is organized. That is to say, in a word, the logical structure of inter-universal Teichmüller theory does not proceed by concatenating some sort of chain of intermediate inequalities. Rather, (∧-Chn) the logical structure of inter-universal Teichmüller theory proceeds by observing a chain of AND relations “∧” [cf. the discussion of [IUTchIII], Remark 3.9.5, (viii), (ix); [IUTchIII], Remark 3.12.2, (c itw ), (f itw ); [Alien], §3.11, (iv), (v)]. As observed in Example 2.4.5, (ii), (iii), once one follows this chain of AND relations “∧” up to and including the multiradial representation of the Θ-pilot [i.e., [IUTchIII], Theorem 3.11], the pas- sage to the final numerical estimate [i.e., [IUTchIII], Corollary 3.12] is relatively straightforward [i.e., as one might expect, from the use of the word “corollary”!]. One essentially formal consequence of (∧-Chn) is the following: Since the defi- nition of the Θ-link, the construction of the multiradial representation of the Θ-pilot, and the ultimate passage to the final numerical estimate consist of a finite number of steps, one natural and effective way to analyze/diagnose [cf. the discussion of §1.4!] the precise content of misunderstandings of inter-universal Teichmüller theory is to determine (∧-Dgns) precisely where in the finite sequence of steps that appear is the first step at which the person feels that the preservation of the crucial AND relator “∧” is no longer clear. In some sense, the starting point of the various AND relations “∧” that appear in the multiradial algorithm of [IUTchIII], Theorem 3.11, is the observation that (∧-Input) the input data for this multiradial algorithm consists solely of an abstract F ×μ -prime-strip; moreover, this multiradial algorithm is functorial with respect to arbitrary isomorphisms between F ×μ -prime- strips [cf. [IUTchIII], Remark 3.11.1, (ii); the final portion of [Alien], §3.7, (i)]. This property (∧-Input) means that the multiradial algorithm may be applied to any F ×μ -prime-strip that appears, or alternatively/equivalently, that any F ×μ - prime-strip may serve as the gluing data [cf. the “γ J in the analogies discussed in §3.2, (StR3), (StR4), as well as Example 2.4.5, (ii)!] between a given situation [i.e., such as the ±ell NF-)Hodge theater in the codomain of the Θ-link!] and the content of the multiradial algorithm. On the other hand, in order to conclude that the multiradial algorithm yields output data satisfying suitable AND relations “∧”, it is necessary also to examine in detail the content of this output data, i.e., in particular, in the context of the central IPL and SHE properties discussed in [IUTchIII], Remark 3.11.1, (iii), as well as the chain of (sub)quotients aspect of the SHE property [cf. [IUTchIII], 106 SHINICHI MOCHIZUKI Remark 3.11.1, (iii); [IUTchIII], Remark 3.9.5, (viii), (ix)]. In a word, the essential “principle” that is applied throughout the various steps of the multiradial algorithm in order to derive new AND relations “∧” from old AND relations “∧” is the following “principle of extension of indeterminacies”: (ExtInd) If A, B, and C are propositions, then it holds [that B =⇒ B C and hence] that A B =⇒ A (B C). One important tool that is frequently used in inter-universal Teichmüller theory in a fashion that is closely related to (ExtInd) is the notion of a poly-morphism [cf. the discussion of §3.7 below for more details]. In the context of (ExtInd), it is interesting to note that, from the point of view of the discussion of §3.4, the “∨” that appears in the conclusion i.e., A (B C) of (ExtInd) may be understood as amounting to essentially the same phenomenon as the “∨” that appears in (NeuORInd2) [e.g., by taking “C” to be A]. That is to say, instead of generating AND relations “∧” tautologically by means of the introduction of distinct labels [i.e., as in (AOΘ1)] i.e., say, by introducing a new distinct label for “C” so as to conclude a tautological relation A B C (ExtInd) allows one to generate new AND relations “∧” while avoiding the introduction of new distinct labels. As discussed in §3.4, this point of view [i.e., of avoiding the introduction of new distinct labels] leads inevitably to OR relations “∨”, i.e., as in (NeuORInd2) or as in the conclusion “A (B C)” of (ExtInd). As discussed above, the reason that one wishes to avoid the introduction of new distinct labels when applying (ExtInd) is precisely that (sQLTL) one wishes to apply (ExtInd) to form “(sub)quotients/splittings” of the log-theta-lattice [cf. the title of [IUTchIII]!], i.e., to project the vertical line on the left-hand side of the infinite “H” portion of the log- theta-lattice onto the vertical line on the right-hand side of this infinite “H” by somehow achieving some sort of “crushing together” of distinct coordinates [i.e., “(n, m)”, where n, m Z] of the log-theta-lattice [cf. the discussion of §3.11 below; [IUTchIII], Remark 3.9.5, (viii), (ix); [IUTchIII], Remark 3.12.2, (c itw ), (f itw ); [Alien], §3.11, (iv), (v)). At this point, it is of interest to note that there are, in some sense, two ways in which (ExtInd) is applied during the execution of the various steps of the multiradial algorithm [cf. the discussion of §3.10, §3.11, below, for more details]: (ExtInd1) operations that consist of simply adding more possibilites/indeter- minacies [which corresponds to passing from B to B C] within some fixed container; (ExtInd2) operations in which one identifies [i.e., “crushes together”, by passing from B to B C] objects with distinct labels, at the cost of passing to LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 107 a situation in which the object is regarded as being only known up to isomorphism. Typical examples of (ExtInd1) include the upper semi-continuity of (Ind3), as well as the passage to holomorphic hulls. Typically, such applications of (ExtInd1) play an important role in establishing various symmetry or invariance properties such as multiradiality. This sort of establishment of various symmetry or invariance prop- erties by means of (ExtInd1) then allows one to apply label crushing operations as in (ExtInd2). Put another way, · (ExtInd1) may be understood as a sort of operation whose purpose is to prepare suitable descent data, while · (ExtInd2) may be thought of as a sort of actual descent operation, i.e., from data that depends on the specification of a member of some collection of distinct labels to data that is independent of such a label specification. [We refer to the discussion of §3.8 below for more details on foundational aspects of (ExtInd2) and to the discussion of §3.9 below for more details concerning the notion of “descent”.] Typical examples of (ExtInd2) in inter-universal Teichmüller theory are the following [cf. the notational conventions of [IUTchI], Definition 3.1, (e), (f)]: · identifying “Π v ”’s [where v V] at different vertical coordinates [i.e., “(n, m)” and “(n, m  )”, for n, m, m  Z] of the log-theta-lattice, which results in a “Π v regarded up to isomorphism” that is labeled by a new label “(n, ◦)”; · identifying “G v ”’s [where v V] at different horizontal or vertical coor- dinates [i.e., “(n, m)” and “(n  , m  )”, for n, n  , m, m  Z] of the log-theta- lattice, which results in a “G v regarded up to isomorphism” that is labeled by a new label “(◦, ◦)”; · identifying the F ×μ -prime-strips in the Θ-link that arise from the Θ- and q-pilot objects in distinct ±ell NF-)Hodge theaters [i.e., the ±ell NF- )Hodge theaters in the domain and codomain of the Θ-link] by working with these F ×μ -prime-strips up to isomorphism. In some sense, the most nontrivial instances of the application of (ExtInd) in the context of the multiradial algorithm occur in relation to the log-Kummer- correspondence [i.e., in the vertical line on the left-hand side of the infinite “H”] and closely related operations of Galois evaluation [cf. the discussion of §3.11 below]. The Kummer theories that appear in this log-Kummer-correspondence i.e., Kummer theories for · multiplicative monoids of nonzero elements of rings of integers in mixed- characteristic local fields, · mono-theta environments/theta monoids, and · pseudo-monoids of κ-coric functions involve the construction of various [Kummer] isomorphisms between · Frobenius-like data and · corresponding data constructed via anabelian algorithms from étale-like objects. 108 SHINICHI MOCHIZUKI The output of such algorithms typically involves constructing the “corresponding data” as one possibility among many. Here, we note that either of these Frobenius-like/étale-like versions of “corresponding data” is unlike, for instance, the data that constitutes an F ×μ -prime-strip! sufficiently robust that it completely determines [even when only regarded up to isomorphism!] the [usual] embedding of the Θ-pilot. That is to say, taken as a whole, the multiradial algorithm and, especially, the portion of the multiradial algorithm that involves the log-Kummer correspondence and closely related operations of Galois evaluation plays the role of exhibiting the Frobenius-like Θ-pilot as one possibility within a collec- tion of possibilities constructed via anabelian algorithms from étale-like data. Thus, in this situation, one obtains the crucial preservation of the AND relation “∧” by applying (ExtInd) twice, i.e., by applying · (ExtInd1) to the enlargement of the collection of possibilities under consideration and · (ExtInd2) to the Kummer isomorphisms involved, when one passes from Frobenius-like object labels “(n, m)” [where n, m Z] to étale-like object labels “(n, ◦)” [where n Z]. This is precisely what is meant by the chain of (sub)quotients aspect of the SHE property [cf. [IUTchIII], Remark 3.11.1, (iii); [IUTchIII], Remark 3.9.5, (viii), (ix)] discussed above [cf. also the discussion of §3.10, §3.11, below]. §3.7. Poly-morphisms and logical AND relations Poly-morphisms i.e., sets of morphisms between objects appear through- out inter-universal Teichmüller theory as a tool for facilitating the explicit enumeration of a collection of possibilities. Composable ordered pairs of poly-morphisms [i.e., pairs for which the domain of the first member in the pair coincides with the codomain of the second member in the pair] may be composed by considering the set of morphisms obtained by composing the morphisms that belong to the sets of morphisms that constitute the given pair of poly-morphisms. Such compositions of poly-morphisms allow one to keep track in a precise and explicit fashion of collections of possibilities under consideration. From the point of view of chains of AND relations “∧”, as discussed in §3.6, the collections of possibilities enumerated by poly-morphisms are to be understood as being related to one another via OR relations “∨”. That is to say, poly-morphisms may be thought of as a sort of indeterminacy, which is used in inter-universal Teichmüller theory to produce structures that satisfy various symmetry or invariance properties, hence yield suitable descent data [cf. the discussion of (ExtInd1) in §3.6; the discussion of §3.9 below]. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 109 Thus, for instance, in the case of the full poly-isomorphism that constitutes the Θ-link, one may understand the fundamental AND relation “∧” of (AOΘ1) which, for simplicity, we denote by A B [where A and B correspond, respectively, in the notation of the discussion of §3.4, to “∗ :→ q-plt Ring” and “∗ :→ Θ-plt Ring”] may be understood as a relation “A (B 1 B 2 . . . )”, i.e., a relation to the effect that if one fixes the q-pilot q-plt, then this q-pilot is glued, via the Θ-link, to the Θ-pilot Θ-plt by means of one isomorphism [of the full poly- isomorphism that constitutes the Θ-link] or another isomorphism, or yet another isomorphism, etc. [Here, the various possible gluings that constitute B are denoted by B 1 , B 2 , . . . .] In particular, as discussed in (∧-Chn), if one starts with the Θ-link and then con- siders various subsequent logical AND relations “∧” that arise for instance, by considering various composites of poly-morphisms! by applying (ExtInd), then (∧(∨)-Chn) the essential logical structure of inter-universal Teichmüller theory, as discussed in (∧-Chn), may be understood as follows: A B = A (B 1 B 2 . . . ) =⇒ A (B 1 B 2 . . . B 1  B 2  . . . ) =⇒ A (B 1 B 2 . . . B 1  B 2  . . . B 1  B 2  . . . ) .. . Finally, we recall that various “classical examples” of the notion of a poly- morphism include · the collection of maps between topological spaces that constitutes a ho- motopy class, or stable homotopy class, of maps; · the collection of morphisms between complexes that constitutes a mor- phism of the associated derived category; · the collection of morphisms obtained by considering some sort of orbit by some sort of group action on the domain or codomain of a given morphism [cf. the discussion of [Rpt2018], §13, (PMEx1), (PMEx2), (PMQut)]. Also, in this context, it is useful to recall [cf. the discussion of [Alien], §4.1, (iv)] that · gluings via poly-morphisms are closely related to the sorts of gluings that occur in the construction of algebraic stacks [i.e., algebraic stacks which are not algebraic spaces]. §3.8. Inter-universality and logical AND relations One fundamental aspect of inter-universal Teichmüller theory lies in the con- sideration of distinct universes that arise naturally when one considers Galois 110 SHINICHI MOCHIZUKI categories i.e., étale fundamental groups associated to various schemes. Here, it is important to note that, when phrased in this way, this fundamental aspect of inter-universal Teichmüller theory is, at least from the point of view of mathematical foundations, no different from the situation that arises in [SGA1]. On the other hand, the fundamental difference between the situation considered in [SGA1] and the situations considered in inter-universal Teichmüller theory lies in the fact that, whereas in [SGA1], the various distinct schemes that appear are related to one another by means of morphisms of schemes or rings, the various distinct schemes that appear in inter-universal Teichmüller theory are related to one another, in general, by means of relations such as the log- and Θ-links that are non-ring/scheme-theoretic in nature, i.e., in the sense that they do not arise from morphisms of schemes or rings. In general, when considering relations between distinct mathematical objects, it is of fundamental importance to specify those mathematical structures that are common i.e., in the terminology of inter-universal Teichmüller theory, coric to the various distinct mathematical objects under consideration. Here, we observe that this notion of being “common”/“coric” to the various distinct mathemat- ical objects under consideration constitutes, when formulated at a formal, symbolic level, a logical AND relation “∧”. cf. the discussion of §3.2 [cf., especially, Example 3.2.2], §3.4, §3.5, §3.6, §3.7. Thus, in the situations considered in [SGA1], the ring/scheme structures of the various distinct schemes that appear are coric and hence allow one to relate the universes/Galois categories/étale fundamental groups associated to these distinct schemes in a way that makes use of the common ring/scheme structures between these schemes. At a concrete level, this means that in the situations considered in [SGA1], étale fundamental groups may be related to one another in such a way that the only indeterminacies that occur are inner automorphism indeterminacies. Moreover, these inner automorphism indeterminacies are by no means superfluous cf. the discussion of Examples 3.8.1, 3.8.2, 3.8.3, 3.8.4 below. Example 3.8.1: Inevitability of inner automorphism indeterminacies. The unavoidability of inner automorphism indeterminacies may be understood in very elementary terms, as follows. def (i) Let k be a perfect field; k an algebraic closure of k; N G k = Gal(k/k) a normal closed subgroup of G k ; σ G k such that the automorphism ι σ : N N of N given by conjugating by σ is not inner. [One verifies immediately that, for instance, if k is a number field or a mixed-characteristic local field, then such N , σ do def indeed exist.] Write k N k for the subfield of N -invariants of k, G k N = N G k , LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 111 def Q N = G k /G k N . Then observe that this situation yields an example of a situation in which one may verify directly that the functoriality of the étale fundamental group only holds if one allows for inner automorphism indeterminacies in the definition of the étale fundamental group. Indeed, let us first observe that the “basepoints” of k and k N determined by k allows us to regard G k and G k N , respectively, as the étale fundamental groups of k and k N . Thus, if one assumes that the functoriality of the étale fundamental group holds even in the absence of inner automorphism indeterminacies, then the commutative diagram of schemes Spec(k N ) σ −→ Spec(k N )   Spec(k) [where the diagonal morphisms are the natural morphisms] induces a commutative diagram of profinite groups G k N ι σ −→  G k N  G k which [since the natural inclusion N = G k N → G k is injective!] implies that ι σ is the identity automorphism, in contradiction to our assumption concerning σ! (ii) The phenomenon discussed in (i) may be understood as a consequence of the fact that, whereas Spec(k) is coric in the commutative diagram of schemes that appears in (i) [i.e., in the sense that this diagram does indeed commute!], Spec(k) is not coric in the diagram of schemes Spec(k)   Spec(k N ) σ −→  Spec(k N )  Spec(k) [where the diagonal morphisms are the natural morphisms], i.e., in the sense that the upper portion of this diagram does not commute! (iii) Finally, we consider the natural exact sequence 1 −→ G k N −→ G k −→ Q N −→ 1 of profinite groups. Then observe that the inner automorphisms indeterminacies of G k [cf. the discussion of (i), (ii)!] induce outer automorphism indeterminacies of G k N that will not, in general, be inner. That is to say, 112 SHINICHI MOCHIZUKI if one considers G k N in the context of this natural exact sequence, then one must in fact consider G k N [not only up to inner automorphism in- determinacies, i.e., as discussed in (i), (ii), but also] up to certain outer automorphism indeterminacies. Relative to the point of view of the discussion of (ii), these outer automorphism indeterminacies may be understood as a consequence of the fact that, in the context of the field extensions k k N k and the automorphisms of these field extensions induced by elements of G k , the field k is coric, whereas the field k N is not coric i.e., in the context of these field extensions and automorphisms of field extensions, the relationship of k to the various field extensions that appear is constant and fixed, whereas the relationship of k N to the various field extensions that appear is variable, i.e., subject to indeterminacies arising from the action of elements of G k . Example 3.8.2: Inter-universality and the structure of ±ell NF-)Hodge theaters. In the following discussion of ±ell NF-)Hodge theaters, we fix a col- lection of initial Θ-data (F /F, X F , l, C K , V, V bad mod , ) as in [IUTchI], Definition 3.1, and apply the notational conventions of [IUTchI], def Definition 3.1. In particular, we recall that E = E F is an elliptic curve over the number field F ; F is an algebraic closure of F ; l is a prime number; K F is the extension field of F determined by the composite of the fields of definition of the closed points of the finite group scheme E[l] E of l-torsion points of E; F mod F is the field of moduli of E, i.e., the field extension of the field of rational numbers generated by the j-invariant of E. For simplicity, we assume that l > 5. (i) We begin by recalling the following: (i-a) The point of view of classical Galois theory with regard to constructing finite Galois extensions of fields may be summarized, in the case of the Galois extension K/F , as follows: · one starts with a base field F ; · one then constructs a finite field extension K of F that is saturated with respect to Galois conjugation over F . Thus, relative to this classical point of view, one is constrained to viewing the situation from the point of view of the base field F . This constraint obliges one to always take into account the entirety of Galois conjugates [over F ] of objects associated to K. The point of view of (i-a) is fundamentally incompatible with the main goal of the construction of ±ell NF-)Hodge theaters in [IUTchI], namely, the simulation of a global multiplicative subspace of E[l] [cf. the discussion of global multi- plicative subspaces in [IUTchI], §I1; [Alien], §2.3, §2.4; [Alien], §3.3, (iv), as well as Example 3.2.1, (vi), of the present paper], together with a global canonical generator, up to ±1, of the quotient of E[l] by the global multiplicative subspace LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 113 [cf. the discussion of global canonical generators in [IUTchI], §I1; [Alien], §3.3, (iv), as well as Example 3.2.1, (vi), of the present paper]. In some sense, the technical starting point of the “simulation of a global multiplicative subspace” implemented in [IUTchI] may be summarized as follows: (i-b) The “simulation of a global multiplicative subspace” given in [IUTchI] is achieved by, in some sense, reversing the flow of the classical construction reviewed in (i-a), i.e., by · viewing the situation [not from the point of view of the base field F , but rather] from the point of view of the hyperbolic orbicurve C K which may be thought of as data that amounts to K, together with a fixed choice of a quotient “Q” [cf. [IUTchI], Definition 3.1, (f)] of E[l], i.e., whose kernel is to serve as the “simulated global multiplicative subspace” and · regarding the base field F mod or, at the level of hyperbolic orbicurves, C F mod [cf. [IUTchI], Remark 3.1.7, (i)] as a finite étale quotient of K [or, at the level of hyperbolic orbicurves, C K ], i.e., which amounts to thinking in terms of [compatible] finite étale quotients Spec(K)  Spec(F mod ), C K C F mod which are regarded as objects constructed from Spec(K), C K . This approach allows one to concentrate on a fixed [simulated global mul- tiplicative] subspace and hence [unlike the situation discussed in (i-a)!] to exclude the various nontrivial Galois conjugates over F of this fixed simulated global multiplicative subspace. The approach of (i-b) has numerous important technical consequences [to be dis- cussed in (ii), (iii), (iv), below]. (ii) From the point of view of étale-like objects [i.e., arithmetic fundamental groups], constructing a quotient C K C F mod as in (i-b) corresponds to constructing a profinite group “Π C F mod from the profinite group Π C K that contains Π C K as an open subgroup. In light of the well-known slimness of Π C F mod [cf., e.g., [AbsTopI], Proposition 2.3, (ii)], such a construction of “Π C F mod amounts to the construction of a finite group of outer automorphisms of some open subgroup of Π C K . This finite group may be thought of as a finite quotient group Π C F mod  Γ mod of Π C F mod . If we think of the absolute Galois group G F mod of the number field F mod as a quotient Π C F mod  G F mod of Π C F mod , then this finite quotient group Γ mod determines a finite quotient group G F mod  Γ Gal mod of G F mod . Here, we recall from the construction of [IUTchI], Example 4.3, (i), that Γ Gal mod has a natural subquotient that may be  × identified with F  l = F l /{±1}, i.e., which corresponds to the multiplicative F l - symmetry of the ±ell NF-)Hodge theater. In particular, Γ Gal mod , hence also Γ mod , is a finite group of order > 2, which implies, by well-known properties of absolute Galois groups of number fields [cf., e.g., [NSW], Theorem 12.1.7] that def 114 SHINICHI MOCHIZUKI (NoSpl) The surjection G F mod  Γ Gal mod of profinite groups does not admit a splitting. Here, we note that [in light of the well-known slimness of G F mod cf., e.g., [Ab- sTopI], Theorem 1.7, (iii)] this non-existence of a splitting may be reformulated as the assertion that the natural outer action of Γ Gal mod on the kernel Ker(G F mod  Gal Gal Γ mod ) does not admit a lifting to an action of Γ mod on Ker(G F mod  Γ Gal mod ), i.e., to an action that is free of inner automorphism indeterminacies. In particular, it follows [a fortiori!] that the natural outer action of Γ mod on Ker(Π C F mod  Γ mod ) does not admit a lifting to an action of Γ mod on Ker(Π C F mod  Γ mod ), i.e., to an action that is free of inner automorphism indeterminacies. That is to say, in sum- mary, the inner automorphism indeterminacies in these natural outer actions are essential and unavoidable. (iii) The existence of the inner automorphism indeterminacies discussed in (ii) implies, in particular, that the permutations of prime-strips in the multiplicative symmetry portion of a ±ell NF-)Hodge theater induced by the F  l -symmetries ±ell of the NF-)Hodge theater necessarily give rise to inner automorphism indeter- minacies in the isomorphisms between the copies of local absolute Galois groups G v [where v V non ] that appear in prime-strips with distinct labels F  l [cf. [IUTchI], Remark 4.5.1, (iii); [IUTchII], Remark 2.5.2, (iii); [IUTchII], Remarks 4.7.2, 4.7.6; [Alien], §3.6, (iii)]. Put another way, (NoSyn) there is no well-defined synchronization between these copies of G v that appear in prime-strips at distinct labels F  l that is free of inner automorphism i.e., conjugacy indeterminacies. In this context, we recall that such a conjugate synchronization is of fundamen- tal importance in inter-universal Teichmüller theory since it is necessary in order to construct the data that appears in the unit group portion of the F ×μ -prime-strip that appears in the domain of the Θ-link, i.e., data that is required to be free of any dependence on the distinct labels F  l . Such a conjugate synchronization is = F l {±1}, achieved by applying the F l -symmetries [where we recall that F l i.e., relative to the natural action of {±1} on F l ] in the additive symmetry portion of the ±ell NF-)Hodge theater under consideration [cf. [IUTchII], Corollary 3.5, (i); [IUTchII], Remark 3.5.2, (iii); [IUTchII], Remark 4.5.3, (i); [IUTchIII], Theorem 1.5, (iii); [IUTchIII], Remark 1.5.1, (i); [Alien], §3.6, (ii)]. Here, we observe that in order to achieve this conjugate synchronization via the F l -symmetry of the various copies of G v that appear in prime-strips with distinct labels, it is of fundamental importance to keep these copies of G v isolated from the absolute Galois groups of number fields that appear in the discussion of (ii) [i.e., since, as observed in (ii), it is precisely the intrinsic structure of these global absolute Galois groups that gives rise to the unwanted inner automorphism/conjugacy indeterminacies!]. This local-global isolation requirement i.e., in effect, the requirement that def (LGIsl) these copies of the local absolute Galois group G v be regarded not as subgroups of some global absolute Galois group, but rather as coric objects that are treated as being independent of any sort of embedding into a global absolute Galois group [cf. [IUTchII], Remark 4.7.6; [Alien], §3.6, (iii)] will have important consequences, as we shall see in the discussion of (iv) below. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 115 (iv) As discussed in (iii), the issue  (SymIsl) of isolating the F l -symmetry from the F l -symmetry in order to achieve conjugate synchronization is one important reason for imposing the local-global isolation requirement (LGIsl) in the context of the construction of ±ell NF-)Hodge theaters. In fact, however, this property (LGIsl) is fundamental to the entire structure of a ±ell NF-)Hodge theater [cf., [IUTchI], Fig. 6.5; [Alien], Fig. 3.8]. That is to say, the issue (SymIsl) may be thought of as being reflected in the gluing along certain collections of prime- strips between the additive and multiplicative symmetry portions of the ±ell NF- )Hodge theater [cf., [IUTchI], Fig. 6.5; [Alien], Fig. 3.8]. In fact, however, (SctNF) even within the multiplicative symmetry portion of a ±ell NF-)Hodge theater, the goal of simulating a global canonical generator requires one to treat the various prime-strips that appear in the multiplicative symmetry portion of the ±ell NF-)Hodge theater as “sections”, in some suitable sense, of the finite étale quotient Spec(K)  Spec(F mod ) a point of view that is fundamentally incompatible with the prime decompo- sition trees of the number fields K, F mod , hence again requires one to impose (LGIsl) [cf. [IUTchI], Remarks 4.3.1, 4.3.2; [Alien], §3.3, (iv)]. On the other hand, let us recall that the ring structure of the nonarchimedean local field that gives rise to G v cannot be reconstructed from the abstract topological group G v [cf. [NSW], the Closing Remark preceding Theorem 12.2.7; [AbsTopIII], §I3; [Alien], Example 2.12.3, (i)]. In particular, once one imposes (LGIsl), the crucial reconstruction of the ring structures of the nonarchimedean local fields that give rise to the various copies of G v where we recall that such ring structures play a fundamental and indispensable role in the definition of the log-link! can only be conducted if one applies the absolute anabelian algorithms of [AbsTopIII], §1, locally at each v V non [not to G v , but rather] to Π v , i.e., one must always regard each coric copy of G v as a “certain quotient” of a corresponding coric copy of Π v . Indeed, from a historical point of view [cf. the discussion of [IUTchI], Remark 4.3.2], it was precisely these local-global isolation aspects i.e., surround- ing (LGIsl), as discussed in (iii) and the present (iv) of the structure of ±ell NF-)Hodge theaters that motivated the author to develop the absolute anabelian algorithms of [AbsTopIII], §1, in the first place! Example 3.8.3: Truncated vs. profinite Kummer theory and compatibil- ity with the p-adic logarithm. In the following discussion, we fix notation as follows: Let k be a finite extension of Q p , for some prime number p; k an algebraic closure of k. Write def · G k = Gal(k/k); · O k for the ring of integers of k, with maximal ideal m k O k ; · O  O k for the multiplicative monoid of nonzero elements of O k ; k · O × O  for the group of invertible elements of O  ; k k k 116 SHINICHI MOCHIZUKI · O ×  O ×μ for the quotient of O × by the subgroup μ of torsion k k k elements [i.e., roots of unity] of O × ; k × · log k : O  k for the p-adic logarithm on O × . k k ×μ ×   O O O k . Let Π  G k be a topological Thus, G k acts naturally on O k k k group equipped with a surjection onto G k , which determines natural actions of Π on O ×μ  O × O  O k . We shall often think of the pair G k  O  or the pair k k k k Π  O  “abstractly” as a pair consisting of an abstract topological monoid [i.e., k O  ] equipped with a continuous action by an abstract topological group [i.e., G k k or Π]. Also, we shall write N ≥1 for the multiplicative monoid of positive natural numbers. (i) The three types of Kummer theory that occur in inter-universal Teichmüller theory [cf. the discussion of Example 3.8.4, (i-a), (i-b), (i-c), below] involve Kum- mer towers that consist of N -th power maps, for N N ≥1 , on the monoids involved. In the case of the pair G k  O  , one observes immediately that such k N -th power maps satisfy the following properties [cf. the discussion of [IUTchII], Remark 3.6.4, (i)], where we assume that N 2: (i-a) the N -th power map O   O  is not a ring homomorphism, i.e., is not k k compatible with the additive structure underlying the ring structure on O  {0}; k (i-b) the N -th power map O   O  is ) G k -equivariant, hence may be k k thought of as inducing on étale-like cyclotomes constructed via anabelian algorithms from G k or Π the isomorphism functorially induced by some at least from an a priori point of view indeterminate automor- phism [cf. (i-a), which implies that G k or Π must be treated as abstract topological groups, that is to say, as opposed to groups of ring/field auto- morphisms, i.e., “Galois groups/arithmetic fundamental groups”; the dis- cussion preceding Example 3.8.1; the discussion following Example 3.8.4 below; the discussion of (vi-c) below] of G k or Π; (i-c) the N -th power map O   O  alters, at least from an a priori point of k k view, cyclotomic rigidity isomorphisms but not synchronizations between collections of cyclotomic rigidity isomorphisms! between étale- like cyclotomes [cf. (i-b)] and [Frobenius-like] cyclotomes arising from the torsion subgroup of O  cf., e.g., the [in general] nontrivial action of k the N -th power map on the n-th roots of unity in O  for n N ≥1 prime k to N . Note that it follows from (i-a) that, if we think of a “basepoint” as a particular “rigid” choice of an algebraic closure that is free of any conjugacy or N -th power map indeterminacies, then whereas (i-d) the p-adic logarithm k O  O × k k log k −→ k O  O × k k yields a precise, well-defined and, in particular, free of any conjugacy or N -th power map indeterminacies! set-theoretic [but not ring-theoretic!] relationship between LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 117 · the basepoint of the [“abstract”] copy of G k  O  in the domain k of log k and · the basepoint of the [“abstract”] copy of G k  O  in the k codomain of log k [where we think of both of these copies of G k  O  as constituent objects k in the respective Kummer towers in the domain/codomain of log k , as discussed above], i.e., a single unified basepoint that is simultaneously valid for both the domain and codomain of log k , (i-e) the map Log k induced by the p-adic logarithm log k on inverse limits of the Kummer tower ×  lim ←− O k lim ←− O k ←− k lim N N N Log k −→ k O  O × k k [where the inverse limits are over N N ≥1 , and we recall that raising to the N -th power on the “O  in the domain of log k corresponds to multiplying k by N on the “k” in the codomain of log k ] only yields a relationship between · the inverse limit basepoint in the domain of Log k and · the basepoint associated to a single constituent Kummer tower object [with a fixed additive structure!] in the codomain of Log k × [i.e., “O × as opposed to “lim ←− O ”]. k N k Note, moreover, that it follows from (i-b), (i-c) that the basepoint shifts that occur as one passes between different constituent Kummer tower objects via various N -th power maps are indeed at least from an a priori point of view substan- tive/nontrivial in the context of cyclotomic rigidity isomorphisms or synchroniza- tions between cyclotomes. (ii) The situation discussed in (i-e) may be understood as a consequence of the fact that (ii-a) the Kummer tower inverse limit “lim ←− [where we omit the subscript N to simplify notation] is biased toward the multiplicative structure of the rings involved i.e., at the expense of the additive structures of these rings [cf. (i-a)] hence fundamentally incompatible with the “juggling/rotation/permutation” of the additive and multiplicative structures that arises from the log-link [cf. the discussion of Example 3.3.2, (iv)]. In the situation of (i-e), we observe, moreover, that the problem of constructing × some sort of “hyper-multiplicative tower” of copies of, say, the lim O k in the domain of Log k that lifts [i.e., relative to Log k ] the [multiplicative] Kummer tower of copies of O × O  k in the codomain of Log k appears to be unrealistically intractable: k k Indeed, compatibility, relative to Log k , with the [multiplicative] Kummer tower 118 SHINICHI MOCHIZUKI in the codomain of Log k would imply that the transition maps of such a “hyper- multiplicative tower” would, at least at a purely formal computational level (!), necessarily be of the form x = exp(log(x)) exp({log(x)} M ) = exp({log(x)} · {log(x)} M −1 ) = x {log(x)} M −1 where M N ≥1 , and the notation “exp(−)” and “log(−)” is intended in a purely formal computational sense (!). On the other hand, (ii-b) it seems difficult to conceive of any sort of natural approach to construct- ing such “hyper-multiplicative transition maps” × × lim ←− O k ←− O k lim M −1 that realize the purely formal computation “x x {log(x)} for × and, moreover, allow one to relate, in some natural way, the x lim O ←− k × [multiplicative] Kummer theory associated to the lim O k in the do- main of the transition map to the corresponding [multiplicative] Kummer × theory associated to the lim O in the codomain of the transition map. k (iii) Note that the conditions imposed on the “hyper-multiplicative transition maps” in (ii-b) are stated in a somewhat rough and imprecise way. Although it is not clear at the time of writing how to make these conditions completely precise, it does, however, seem natural to consider the possibility of the existence of commutative diagrams as in (iii-a) below, i.e., where one thinks of “Z” as a sort of candidate for the inverse limit of the “hyper-multiplicative transition maps” of (ii-b). That is to say, the existence of such a commutative diagram may be thought of as a sort of necessary condition for the existence of a suitable system of “hyper-multiplicative transition maps” as in (ii-b). [Here, we note that the surjectivity condition of (iii- a) below may be understood as a sort of “very weak necessary” version of the domain/codomain Kummer theory-relatability condition of (ii-b).] In fact, however, we observe that, in the situation of (ii-b) [and the surrounding discussion], such a commutative diagram does not exist: × (iii-a) Let Z be a set with an action by G k , ζ : Z lim ←− O k a G k -equivariant map of sets that induces a surjection from the J-invariants in the domain of ζ to the J-invariants in the codomain of ζ for every closed subgroup J G k that acts trivially on μ k. [Thus, ζ itself is necessarily surjective.] Then there does not exist any commutative diagram of the form λ −→ lim k Z ←− ζ ψ Log k × lim k ←− O k −→ where λ is a G k -equivariant map of sets, and ψ is the natural projection. (iii-b) The map Log k does not admit any factorization × lim ←− O k N λ −→ lim ←− k N ψ −→ k LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 119 where λ is a G k -equivariant map of sets, and ψ is the natural projection. Indeed, since (iii-b) follows formally from (iii-a), it suffices to verify (iii-a). Suppose that a commutative diagram as in (iii-a) exists. Write · O k = O k k, O k × = O × k, m k = m k k; def def def k def · F = k(μ ) k. Let x p 2 · O k be a nonzero element. Write · f = 1 + x O k × , (0  =) y = log k (f ) m k ; · E k for the field extension of F obtained by adjoining all N -th roots of f , for N N ≥1 , in k. def def Thus, f lifts [via the natural projection] to an element of the codomain of ζ that def is fixed by the action of J = Gal(k/E) G k , hence also [by our surjectivity assumption on ζ] to an element f Z Z of the domain of ζ that is fixed by the action of J. Since λ is G k -equivariant, we thus conclude that λ(f Z ) is fixed by the action of J and maps via the natural projection ψ to an element z E k that defines a divisible, hence l-divisible element of E × , for any prime number l  = p. On the other hand, it follows immediately from the commutativity of the diagram that y = z. Next, observe that since [the unit!] f is already clearly l-divisible in k × , hence also in F × , the Galois group Gal(E/F ) is isomorphic to a quotient of Z p . But this implies that all l-power roots of [the non-unit!] y = z k × F × are contained in F , in contradiction to the easily verified fact that the value group of the valued field F is isomorphic to Z[p −1 ]. This completes the proof of (iii-a). (iv) The qualitatively different behavior that occurs in (i-d) and (i-e) may be understood as being a consequence of the fact [cf. (i-a)] that whereas (iv-a) the ring structure on O  {0} [which makes it possible to define k the well-known power series for the p-adic logarithm log k ] remains intact at any particular constituent Kummer tower object [cf. the situation of (i-d)], (iv-b) the natural multiplicative structure on ×  lim ←− O k lim ←− O k ←− k lim N N N does not admit any corresponding additive structure that gives rise to a ring structure [i.e., that would make it possible to define the well- known power series for the logarithm, hence a factorization as in (iii-b)] on any of the three inverse limits in the above display that is compatible with the natural action by G k and the various natural projections to k [cf. the situation of (i-e)]. Moreover, we observe that (iv-c) the various natural G k -equivariant projections of multiplicative monoids lim ←− k  k; N   lim ←− O k  O k ; N × × lim ←− O k  O k N 120 SHINICHI MOCHIZUKI do not admit splittings [as may be seen, for instance, by restricting such a splitting to the roots of unity, where the existence of such a splitting would amount, in particular, to a splitting of the natural surjection Q p  Q p /Z p ]. That is to say, there is no natural way to relate the finite Kummer theory for a single constituent Kummer tower object “k”, “O  ”, “O × in the codomain of the map k k Log k of (i-e) to the corresponding profinite Kummer theory obtained by passing to the inverse limit “lim ←− of the associated Kummer tower. N (v) In the context of (iv-b) [and the surrounding discussion], it is also of interest to observe that in fact (v-a) neither of the inverse limits  def def lim I k = k; I O × = lim ←− k N N O × k  {0} admits a field structure that is stabilized by the natural action of G k , and whose underlying multiplicative structure is the natural multiplicative structure on the inverse limit. Indeed, suppose that I {I k , I O × } admits such a field structure. Let l be an odd k prime that does not divide the order of the finite group of roots of unity of k. Write μ l k for the group of l-th roots of unity of k, l ⊆) μ l k for the group of def def def l-power roots of unity of k, K = k(μ l ) k, G k G K = Gal(k/K), G K/k = def G k /G K , L = I G K [i.e., the subfield of G K -invariants of I]. Then our assumption on l implies that the natural faithful action of G K/k on μ l [which allows us to think of G K/k as a closed subgroup of Z × l ] induces a nontrivial action of G K/k on × μ l , hence [in light of the well-known structure of the profinite group Z × l = Z l × F l for odd primes l] that G K/k contains a nontrivial finite closed subgroup H G K/k . Next, observe that the group of divisible elements of the multiplicative module K × is equal to μ l [cf. the fact that G K/k is isomorphic to a closed subgroup of Z × l ; [Tsjm], Lemma D, (iii), (iv)]. This implies that the multiplicative G K/k -module def L × is naturally isomorphic to the G K/k -module M l = Hom(Q l /Z l , μ l ) Z l Q l [where we note that as an abstract module, M l is isomorphic to Q l ], hence, in particular, that the field L is of infinite cardinality. On the other hand, it follows from elementary Galois theory that L is a finite Galois extension of the subfield L H L of H-invariants of L. Moreover, since H acts nontrivially on μ l , hence also nontrivially on M l , we thus conclude from the corresponding fact for the action × of nontrivial subgroups of the group of Teichmüller representatives [F × l ] Z l on Q l that L H = {0, 1} is a set of cardinality two, hence that the infinite field L is a finite field, a contradiction. This completes the proof of (v-a). Note that (v-b) if it was indeed the case that I O × admits a topological field structure as k in (v-a), then it would be possible to consider the well-known power series for the logarithm [cf. (iv-a), (iv-b)]. Of course, it follows from (v-a) that such a topological field structure does not exist. In particular, the content of (v-a) may be understood as pointing roughly in the same direction as (iii-a), (iii-b), (iv-a), (iv-b). LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 121 (vi) Thus, in summary, (vi-a) the fundamental dichotomy, in the context of the p-adic logarithm, between · [finitely] truncated Kummer theory [as in (i-d)] and · profinite Kummer theory [as in (i-e)] may be understood in terms of the existence [cf. the “k” in the domain of log k in (i-d)] versus non-existence [cf. the “correspondence” (Z, ζ, λ) of (iii-a)] of a single unified basepoint for the Kummer theories in the domain/codomain of log k or Log k , i.e., a single set equipped with an action by G k that is “sufficiently rich” as to admit subquotients [i.e., where we think of λ as in (iii-a) as being surjective] that correspond to the Kummer theories in the domain/codomain of log k or Log k . That is to say, (vi-b) in the case of profinite Kummer theory, the non-existence of such a single unified basepoint means that one must treat the Kummer theories in the domain/codomain of Log k i.e., at a more concrete level, the sets I k or I O × equipped with their natural multiplicative structures, profinite k cyclotomes, and G k -actions as being independent of one another. In particular, it follows, essentially formally, from (vi-b) that (vi-c) one must think of the copies of “G k that appear in the Kummer theories in the domain/codomain of the p-adic logarithm which may in fact arise as quotients of copies of some topological group Π in the domain/codomain of the p-adic logarithm as being related to one another not as groups of automorphisms of the various monoids that appear in the Kummer theories in the domain/codomain of the p-adic logarithm, but rather as abstract topological groups, i.e., which may be related to one another only by means of some indeterminate isomorphism of topological groups Π Π cf. the discussion preceding Example 3.8.1, as well as the discus- sion following Example 3.8.4, concerning the necessity of working, in the context of the log- and Θ-links, with abstract topological groups, that is to say, as opposed to groups of ring/field automorphisms, i.e., “Galois groups/arithmetic fundamental groups”. Here, we observe that the isomorphism indeterminacy discussed in (vi-c) includes, in particular, inner automorphisms of Π. This chain of observations (vi-a), (vi-b), (vi-c) forms the starting point of the discussion of Example 3.8.4 below. Example 3.8.4: Symmetrizing isomorphisms, truncatibility, and the log- Kummer- correspondence. We maintain the notation of Examples 3.8.2, 3.8.3. 122 SHINICHI MOCHIZUKI Also, we shall write “Out(−)” for the group of outer automorphisms [i.e., arbi- trary automorphisms considered up to inner automorphisms] of a topological group “(−)”. (i) In the following discussion, we consider the issue of compatibility between · the various symmetrizing isomorphisms arising from the action of the  F  l - and F l -symmetries on [“abstract”] copies of the pair G k  O k [cf. the discussion of Examples 3.8.2, 3.8.3; the theory of [IUTchI], §4, §5, §6; [Alien], §3.3, (v); [Alien], §3.6, (i), (ii), (iii)] and · the log-Kummer-correspondence ... ...   ... ... [cf. [IUTchIII], Theorem 3.11, (ii)] for the three types of Kummer theory, namely, (i-a) the Kummer theory for [“abstract”] copies of the pair G k  O  , which is k based on the classical theory of Brauer groups/local class field theory for p-adic local fields [cf. [Alien], §3.4, (v)]; (i-b) the Kummer theory surrounding theta functions and theta values [cf. [Alien], §3.4, (iii), (iv); [Alien], §3.6, (ii)]; (i-c) the Kummer theory surrounding κ-coric functions and copies of the number field F mod [cf. the discussion of Example 3.8.2; [IUTchI], Defi- nition 3.1, (b); [Alien], §3.4, (ii); [Alien], §3.6, (iii)]] that appear in inter-universal Teichmüller theory [cf., e.g., [Alien], Fig. 3.10]. Be- fore proceeding, we recall that the symmetrizing isomorphisms arising from the  action of the F  l - and F l -symmetries on [“abstract”] copies of the pair G k  O k differ in that · whereas the symmetrizing isomorphisms arising from the F l -symmetries are free of inner automorphism indeterminacies and hence give rise to conjugate synchronizations, · the symmetrizing isomorphisms arising from the F  l -symmetries neces- sarily involve inner automorphism indeterminacies [cf. the discussion of Example 3.8.2]. On the other hand, it follows immediately i.e., by considering the symmetrizing isomorphisms induced by arbitrary elements of Aut(X K ) Out(Π X K ) [cf. the notation of [IUTchI], Definition 3.1, (d); [Alien], §3.3, (v)] that if one forgets about the issue of conjugate synchronization and just thinks in terms of arbitrary [indeterminate] isomorphisms between [“abstract”] copies of the pair G k  O  , then the symmetrizing isomorphisms arising k from the action of the F  - and F l l -symmetries on [“abstract”] copies of the pair G k  O  in fact coincide. k LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 123 (ii) In the case of the Kummer theory of (i-a), (ii-a) compatibility between the F l -symmetrizing isomorphisms i.e., without conjugacy indeterminacies! [cf. the final portion of (i)] and the Kummer theories of (i-a) in the domain/codomain of the log- link then follows formally by applying transport of structure via the F l - symmetries to the truncated Kummer theories in the domain/codomain of the log-link, computed relative to the single unified basepoint dis- cussed in Example 3.8.3, (i-d), (vi-a), at each evaluation label “t F l [cf. the discussion of [IUTchIII], Remark 2.3.3, (viii); [Alien], §3.6, (ii)]. Here, we recall that (ii-b) this compatibility is necessary to define the diagonal “0/  / >” i.e., with respect to the evaluation labels “t F l local data that is used to construct the local unit group [i.e., “O ×μ ”] portion of the gluing data that appears in the Θ-link [cf. [IUTchIII], Theorem 1.5, (iii); [IUTchIII], Remark 1.5.1, (i); [Alien], §3.6, (ii)]. In this context, it is important to recall that (ii-c) this local unit group portion of the Θ-link gluing data satisfies the crucial property of being independent of the evaluation labels “t F l which are only well-defined internally within a particular ±ell NF-)Hodge the- ater! hence allows one to construct the containers [that is to say, in the form of tensor-packets of log-shells] that appear in the multiradial representation of the Θ-pilot, i.e., the containers that make it possible to represent the Θ-pilot in the domain ±ell NF-)Hodge theater of the Θ-link in terms of “external” data arising from the codomain ±ell NF- )Hodge theater of the Θ-link. Put another way, (ii-d) if it were the case that the containers of (ii-c) could only be constructed from the domain ±ell NF-)Hodge theater of the Θ-link in a way that involves independent conjugacy indeterminacies · at the distinct evaluation labels “t F l [cf. (ii-b), (ii-c)] or · in the domain/codomain of the log-link [cf. (ii-a)] i.e., all of which are only well-defined internally within the domain ±ell NF-)Hodge theater of the Θ-link! then it would follow that these containers cannot be constructed in a way that is well-defined externally to the domain ±ell NF-)Hodge theater of the Θ-link, e.g., in terms of data arising from the codomain ±ell NF-)Hodge theater of the Θ-link. In particular, we observe that the truncatibility of the Kummer theory of (i- a) plays a fundamental role in the logical structure of inter-universal Teichmüller theory. (iii) The discussion of (ii-b), (ii-c), (ii-d) centers on the issue of synchro- nizing the conjugacy indeterminacies at the diagonal label “ / >” in the do- main/codomain of the log-link. This focus of attention on the diagonal label thus prompts the following question: 124 SHINICHI MOCHIZUKI (iii-a) If one is only interested in synchronizing the conjugacy indeterminacies at the diagonal label “ / >” in the domain/codomain of the log-link, then why does it not suffice to relate the profinite [i.e., rather than truncated!] versions of the Kummer theory of (i-a) for the diagonal label “ / >” in the domain/codomain of the log-link via a single [i.e., corresponding to the “single” diagonal label] indeterminate isomorphism “Π Π” as in Example 3.8.3, (vi-c)? In fact, however, the approach described in (iii-a) is not sufficient for the following reason: (iii-b) Ultimately, in inter-universal Teichmüller theory, one is interested in con- structing the multiradial representation of the Θ-pilot [cf. [IUTchIII], Theorem 3.11] which involves the theta values 2 “q j v at v V bad i.e., whose construction depends, in an essential way, on the use of [global] independent labels “t F l [cf. (ii-d)] at which the conjugacy indeterminacies are nevertheless synchronized relative to a single basepoint arising from the global additive symmetry portion “D of the ±ell NF-)Hodge theater under consideration [cf. [IUTchI], Definition 6.1, (v); [IUTchII], Corollary 4.5, (iii), (iv); [IUTchII], Corollary 4.6, (iii), (iv)]. That is to say, (iii-c) the requirements discussed in (iii-b) are satisfied by the approach that is actually taken in inter-universal Teichmüller theory [cf. (ii-a); [IUTchIII], Proposition 1.3, (i); [IUTchIII], Remark 1.3.2], i.e., of synchronizing the conjugacy indeterminacies in the domain/codomain of the log-link locally “label by label”, for the various labels “t F l [cf. (ii-d)], so that arbitrary conjugacy indeterminacy synchronizations in the domain of the log-link are reflected faithfully in the codomain of the log-link. On the other hand, (iii-d) the approach discussed in (iii-c) can be implemented precisely because of the existence of the canonical single unified basepoints for the do- main/codomain of the log-link discussed in Example 3.8.3, (vi-a), i.e., which are available only in the case of truncated Kummer theory, since the profinite Kummer theory conjugacy indeterminacies of Example 3.8.3, (vi-c), would give rise [in the situation of the approach discussed in (iii-c)] to independent conjugacy indeterminacies at the various labels “t F l ”. Thus, in summary, the discussion of the present (iii) sheds further light on the fundamental role played by the truncatibility of the Kummer theory of (i-a) in the logical structure of inter-universal Teichmüller theory. (iv) In the case of the Kummer theory/Galois evaluation of (i-c), let us first recall from the discussion of [IUTchIII], Remark 2.3.3, (vi), (vii) [cf. also [Alien], §3.4, (ii)] that LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 125 (iv-a) the fact that the submonoid I ord N × {±1} generated by the set of orders of the zeroes/poles [considered as signed elements of N × {±1}] of the rational functions that appear in (i-c) contains i.e., unlike the case with the theta functions that appear in (i-b)! elements ∈ {±1}, as well as elements ∈ N, means that the cyclo- tomic rigidity isomorphisms obtained in (i-c) may only be constructed in a fashion consistent with the anabelian reconstruction algorithms of [AbsTopIII], Theorem 1.9 [cf. also [IUTchI], Remark 3.1.2, (ii), (iii)] if one constructs these cyclotomic rigidity isomorphisms · via profinite Kummer theory [i.e., by applying the fact that  × = {1}] and Q >0 Z · up to an indeterminacy given by multiplication by elements ord  {±1} to the of the image I ord ± = {±1} of the projection I second factor. i.e., in particular, relative to the constraints discussed in Example 3.8.3, (vi-b), (vi-c). That is to say, it follows from the discussion of Example 3.8.3, (vi-b), (vi-c) cf. also · the splitting/decoupling of the unit group portion from the pseudo- monoid of κ-coric rational functions [as discussed in [IUTchI], Example 5.1, (v); [Alien], §3.4, (ii)]; · the non-interference properties satisfied by the Frobenius-like copies of × in the domain/codomain of the log-link [as discussed in [IUTchIII], F mod Proposition 3.10, (ii); [Alien], §3.7, (i)] that (iv-b) the Kummer theories/Galois evaluation operations of (i-c) in the do- main/codomain of the log-link · involve [pseudo-]monoids that must be treated indepen- dently of one another, hence, in particular, · may be related to one another only up to indeterminacies that involve indeterminate isomorphisms between corresponding Ga- lois groups/arithmetic fundamental groups in the domain/codomain of the log-link [cf., especially, the discussion of the final portion of Example 3.8.3, (vi)]. On the other hand, (iv-c) the compatibility between the F  l -symmetrizing isomorphisms i.e., with conjugacy indeterminacies! [cf. the discussion of the final portion of (i)] and the Kummer theories/Galois evaluation operations of (i-c) in the domain/codomain of the log-link follows formally by applying transport of structure via the F  l -symmetries to the profinite Kummer 126 SHINICHI MOCHIZUKI theories of (i-c) in the domain/codomain of the log-link, together with the compatibility of the log-link with these F  l -symmetrizing isomorphisms [cf. [IUTchIII], Proposition 1.3, (i), (ii); [IUTchIII], Remark 1.3.3, (ii)]. As a result of the various indeterminacies of (iv-b), (iv-c), (iv-d) in the case of the Kummer theories/Galois evaluation operations of (i- c), the only diagonal “0/  / >” i.e., with respect to the evaluation labels “j F  l number field inside the algebraic closure F that is well-defined externally to the domain ±ell NF-)Hodge theater of the Θ-link [cf. the discussion of the multiradial representation in (ii-c), (ii-d)] is F mod [cf. [Alien], §3.6, (iii)]. Here, we observe that (iv-e) the I ord ± -indeterminacies of (iv-a) may be synchronized · relative to the F  l -symmetry by applying the “linear dis- jointness” of the theory of κ-coric functions from SL 2 (F l ) [cf. the second display of [Alien], §3.6, (iii)]; · relative to the distinct valuations v V in local-global com- parisons of the Kummer theories/Galois evaluation operations of (i-c) by means of comparison with the cyclotomic rigidity iso- morphisms arising from the Kummer theories of (i-a) [i.e., which are not subject to I ord ± -indeterminacies] [cf. the discussion of the final portion of [IUTchIII], Remark 2.3.3, (vi)], but · not relative to the domain/codomain of the log-link since the relevant [Frobenius-like] monoids in the domain/codomain of the log-link are not set-theoretically related to one another via the log-link [cf. (iv-b)]. In particular, we note that one fundamental aspect of the Kummer theory/Galois evaluation of (i-c) that underlies the compatibility and synchronization properties of (iv-c), (iv-e) is the F  l -symmetricity of the Kummer theory/Galois evaluation of (i-c) [cf. the discussion of [IUTchII], Remark 1.1.1, (v); the first display of [IUTchIII], Remark 2.3.3, (iii); the discussion of [Alien], §3.6, (iii)]. (v) The Kummer theory/Galois evaluation of (i-b), which does not satisfy the condition of F l -symmetricity, exhibits qualitatively fundamentally different behavior from the F  l -symmetric Kummer theory/Galois evaluation of (i-c) [cf. the discussion of [IUTchII], Remark 1.1.1, (v); the first display of [IUTchIII], Remark 2.3.3, (iii)]. If, for instance, (v-a) one attempts to apply the approach via profinite Kummer theory of (iv-b), (iv-c) to the task of verifying some sort of compatibility be- tween the F l -symmetrizing isomorphisms i.e., with conjugacy indeterminacies arising from log-link domain/codomain comparisons, as discussed in Example 3.8.3, (vi-b), (vi-c) [cf. also the discussion of (i), (ii), (iii), especially (iii-d), of the present Example 3.8.4]! and the Kummer LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY theories/Galois evaluation operations of (i-b) in the domain/codomain of the log-link, then one encounters the following situation: (v-b) In order to compare, via the F l -symmetry, the Kummer theories/Galois evaluation operations at different t F l [i.e., for distinct theta values 2 “q j ”, where v V bad ], it is necessary to establish a(n) a priori self- v contradictory!— situation in which the F l -symmetry permutes the labels “t F l in a nontrivial fashion, but acts trivially on the [non- F l -symmetric!] étale theta function! Here, we recall [cf. the theory of [IUTchII], §2] that (v-c) in the context of Galois evaluation, the étale theta function may be thought of as a cohomology class   some closed subgroup some cyclotome associated to the H 1 , of Π v Galois evaluation label t F l that is to say, where [cf. (v-b)!] the F l -symmetry is to act nontriv- ially on the Galois evaluation cyclotome [i.e., the second argument of “H 1 (−)”], but trivially on the closed subgroup of Π v [i.e., the first argument of “H 1 (−)”]! Before proceeding, it is also of interest to recall that (v-d) a profinite Kummer theory situation of the sort discussed in (v-a), (v-b), (v-c) occurs, for instance, if one replaces the theta functions that occur in inter-universal Teichmüller theory by their N -th powers, where N is an integer 2 [cf. the discussion of [IUTchIII], Remark 2.3.3, (vi), (vii)]. Now returning to the discussion of (v-b), (v-c), we observe that (v-e) it is essentially a tautology that the only “solution” to the [a priori] self-contradictory simultaneous “trivial/nontrivial action” condi- tion “(SimCon)” of (v-b), (v-c) lies in working with objects that are invariant with respect to the F l -symmetry, i.e., at a more concrete level, with a single unified set-theoretic basepoint which allows one to compute the various F l -symmetrizing isomor- phisms by projecting to the single copy of “G v determined by the base- point. On the other hand, (v-f) the existence of the conjugacy indeterminacies inherent in the profi- nite Kummer theory F l -symmetrizing isomorphisms of (v-a) means that any “single unified basepoint” as in (v-e) is well-defined only up to conju- gacy indeterminacies a situation that is unacceptable in inter-universal Teichmüller theory since it would mean that the coefficient cyclotomes in  × -multiples, i.e., that the étale (v-c) are well-defined only up to certain Z 127 128 SHINICHI MOCHIZUKI 2 theta functions and theta values [i.e., “q j ”, where v V bad ] in the v  × -powers [cf. the discussion theory are well-defined only up to certain Z of [IUTchIII], Remark 2.1.1, (v)]! That is to say, (v-g) the only way to avoid the pathologies of (v-f) is to replace the profinite Kummer theory in the discussion of (v-a), (v-b), (v-c), (v-e), (v-f) by the truncated Kummer theory of (i-b) [cf. the discussion of the final por- tion of [Alien], §3.6, (ii)], so that we can apply the single unified basepoint of Example 3.8.3, (i-d), (vi-a) [cf. also the discussion of (ii) in the present Example 3.8.4] to obtain a single unified set-theoretic basepoint as in (v-e), but which is well-defined up to geometric fundamental group conjugacy indeterminacies [i.e., indeterminacies arising from conjugation by elements of the geometric fundametric fundamental groups involved], which are, at any rate, inherent in the F l -symmetrizing iso- morphisms and, moreover, [unlike the situation discussed in (v-f)!] do not have any effect on the computation of these F l -symmetrizing iso- morphisms by projecting to the single copy of “G v determined by the basepoint cf. [IUTchII], Remark 1.1.1, (iv), (v); [IUTchII], Remark 2.6.1, (i), (ii). Before proceeding, it is interesting to observe that (v-h) the “tautological resolution” of the [a priori] self-contradictory simul- taneity condition (SimCon) discussed in (v-e), (v-g) by considering “invariants” i.e., in the situation of (v-e), (v-g), a single unified set- theoretic basepoint is formally highly reminiscent of the “tautological resolution” of the log-shifts in the left- and right-hand columns of the “infinite H” of (InfH) by means of the construction of invariants [cf. the discussion surrounding (logORInd), (Di/NDi) in §3.11 below; the discus- sion of (Stp7) in §3.10 below]. Thus, in summary, the truncatibility of the Kummer theory/Galois evaluation operations of (i-b) [cf. the discussion of [IUTchIII], Remark 2.3.3, (vi), (vii), (viii); the final portion of [Alien], §3.6, (ii)] plays a fundamental role in the logical structure of inter-universal Teichmüller theory [cf. the discussion of Example 3.3.2, (vii)]. (vi) Recall that the single unified basepoint of (v-g) is completely deter- mined by the connected subgraph “Γ  X Γ X ”, or, equivalently, the connected Γ [cf. [IUTchII], Remark 2.6.1, (i), (ii); [IUTchII], Remark subgraph “Γ  2.6.3, (i)]. Here, we recall further from [IUTchII], Remark 2.6.3, (i), that Γ X or Γ may be thought of as a “copy Γ of the real line R”, in which the integers Z R are taken to be the vertices, and the line segments joining the integers are taken to be the edges. In light of the central role played by the singled unified basepoint of (v-g) in the discussion of (v) and indeed in the entire logical structure of inter-universal Teichmüller theory! it is of interest to recall from the discus- sion of [IUTchII], Remark 2.6.3, that this subgraph “Γ  is [essentially] uniquely LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 129 determined by various natural conditions, which must be satisfied in order for the theory to operate in the desired fashion. Indeed, let us first recall from the discussion of [IUTchII], Remark 2.5.2, (i), (ii), (iii), (iv); [IUTchII], Remark 2.6.3, (ii), that (vi-a) the various geometric fundamental groups “Δ” that appear act on the various vertices t {−l  , . . . , −1, 0, 1, . . . , l  } Γ independently where we recall that l  = l−1 2 , and that (vi-b) the actions referred to in (vi-a) of geometric fundamental groups “Δ” on the “copy Γ of the real line R” amount to the conventional action of def G Γ = l · Z  {±1} [i.e., the group generated by translations by elements of l · Z and multiplication by −1] on the real line R. def  Z; Write Γ  Z = Γ def   ±Γ  Z = Γ Z −Γ Z R. Next, recall that (vi-c) since the vertex 0 Γ is the unique vertex in Γ that may be used to define the crucial splittings of theta monoids discussed in [IUTchII], Corollary 2.6, (ii), it is necessary that 0 Γ  Z [cf. [IUTchII], Remark 2.6.3, (i)]. On the other hand, (vi-c), together with the existence of the independent G Γ - indeterminacies of (vi-a), (vi-b), imply that (vi-d) if Γ  contains a connected component Γ  that is not contained in l · Z (⊆ Γ), then any theta value 2 q j v obtained via Galois evaluation for j Γ  Z Γ, is well-defined only up to a G Γ -indeterminacy, i.e., up to a possible confusion between j and G Γ -translates j  of j. (vi-e) any theta value q j 2 v obtained via Galois evaluation for j Γ  Z , is well-defined only up to , i.e., up to a possible confusion between j a G Γ -indeterminacy within Γ  Z   and G Γ -translates j of j that lie inside Γ Z . Here, it is important to recall [cf. [IUTchI], Remark 3.5.1, (ii); [IUTchII], Remark 2.6.3, (iv); [IUTchII], Corollary 4.5, (v); [IUTchII], Corollary 4.6, (v)] that (vi-f) any indeterminacies concerning “j”, “j  of the sort described in (vi-d), (vi-e) for nonzero j, j  Z with distinct absolute values would result in indeterminacies in the “vector of ratios” [cf. [IUTchII], Corollary 4.5, (v)] that determines the structure of the global realified Gaussian Frobenioids, i.e., would result in violations of the global product formula relating the value groups at different v V. In particular, (vi-c), (vi-d), (vi-e), (vi-f) imply that (vi-g) the natural map ±Γ  Z /{±1} Z/G Γ 130 SHINICHI MOCHIZUKI i.e., from the set of {±1}-orbits of ±Γ  Z to the set of G Γ -orbits in Z is injective [cf. [IUTchII], Remark 2.6.3, (ii)]; (vi-h) the subgraph Γ  Γ is connected [cf. [IUTchII], Remark 2.6.3, (i)]. Now one verifies immediately that (vi-c), (vi-g), (vi-h) imply formally that (vi-i) Γ  Γ is a connected subgraph that contains 0 and is contained in the closed interval [−l  , l  ]. Next, we observe that, in light of (vi-e), (vi-i),  (vi-j) replacing Γ  Z by ±Γ Z does not have any effect on the validity of (vi-i) or on the way in which the subgraph Γ  Z is used in inter-universal Teichmüller theory; in particular, we may assume, without loss of generality, that the  symmetry condition Γ  Z = ±Γ Z holds. On the other hand, (vi-k) the estimates that are ultimately obtained in inter-universal Teichmüller theory [cf. [IUTchIII], Corollary 3.12; [IUTchIV], Theorem 1.10] are op- timized precisely when the average of the squares j 2 of the nonzero ele-  ments j Γ  Z = ±Γ Z is maximized [cf. [IUTchII], Remark 2.6.3, (ii)],   i.e., when Γ  = ±Γ  Z is given precisely by the closed interval [−l , l ] [which implies that the natural map of (vi-g) is bijective].  That is to say, in summary, when subject to the symmetry condition Γ  Z = ±Γ Z of (vi-j) and the optimization condition of (vi-k), the subgraph Γ  Γ is in fact uniquely determined [cf. [IUTchII], Remark 2.6.3, (v)]. Unlike the situations considered in [SGA1] [cf. the discussion of Example 3.8.1], in which the ring/scheme structures of the various distinct schemes that appear are coric, the ring structures of the rings that appear on either side of the log- and Θ-links of inter-universal Teichmüller theory i.e., such as number fields or completions of number fields at various valuations are not coric with respect to the respective links. This leads one naturally to consider weaker structures [cf. the discussion of Example 3.2.2, (i), (ii), (iv)] the discussion of Example 3.8.2, (iii), (iv); the discussion of Example 3.8.3, (vi)] such as · abstract topological groups, in the case of the profinite Kummer theories in the domain/codomain of the log-link, or · sets equipped with a topology and a continuous action of a topological group, in the case of the log-link, or · realified Frobenioids [in the sense of [IUTchIII], Theorem 1.5, (v)] or topological monoids equipped with a continuous action of a topological group, in the case of the Θ-link, which are indeed coric with respect to the respective links. Indeed, it is pre- cisely this sort of consideration i.e., of weaker coric structures to relate the uni- verses/Galois categories/étale fundamental groups associated to ring/scheme struc- tures on opposite sides of the links under consideration [cf. the discussion preceding Example 3.8.1] that gave rise to the term “inter-universal”. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 131 Here, we note that it is of fundamental importance that these topological groups [which typically in fact arise as Galois groups or arithmetic fundamental groups of schemes] be treated as abstract topological groups, rather than as Galois groups or arithmetic fundamental groups [cf. the discussion at the beginning of §3.2; the discussion of Example 3.8.2, (iii), (iv); the discussion of Example 3.8.3, (vi)]. That is to say, to treat these topological groups as Galois groups or arithmetic fundamental groups requires the use of the ring/scheme structures involved, i.e., the use of structures which are not available since they are not common/coric to the rings/schemes that appear on opposite sides of the log-/Θ-link [cf. the discussion of [Alien], §2.10; [IUTchIII], Remarks 1.1.2, 1.2.4, 1.2.5; [IUTchIV], Remarks 3.6.1, 3.6.2, 3.6.3]. In this context, it is also of fundamental importance to observe that it is precisely because these topological groups must be treated as abstract topological groups that anabelian results play a central role in inter-universal Teichmüller theory. One consequence of the constraint [discussed above] that one must typically work, in inter-universal Teichmüller theory, with structures that are substantially weaker than ring structures is the necessity, in inter-universal Teichmüller theory, of allowing for various indeterminacies, such as (Ind1), (Ind2), (Ind3), that are somewhat more involved than the relatively simple inner automorphism indetermi- nacies that occur in [SGA1]. Here, we recall that from the discussion of (∧(∨)-Chn) in §3.7 that it is precisely the numerous indeterminacies that arise in inter-universal Teichmüller theory that give rise to the numerous logical OR relations “∨” in the display of (∧(∨)-Chn). On the other hand, once one takes such indeterminacies into account, i.e., once one consents to work with various objects “up to certain suitable indeterminacies” e.g., by means of poly-morphisms, as discussed in §3.7 it is natural to identify, by applying (ExtInd2) [as discussed in §3.6], objects that are related to one another by means of collections of isomorphisms [i.e., poly-isomorphisms] that are uniquely determined up to suitable indeterminacies. Here, we observe that this sort of (ExtInd2) identification that occurs re- peatedly in inter-universal Teichmüller theory [cf. the discussion of §3.6] may at first glance appear somewhat novel. In fact, however, from the point of view of mathematical foundations i.e., just as in the discussion of inter-universality given above! this sort of (ExtInd2) identification is qualitatively very similar to nu- merous classical constructions such as the following: (AlgCl) the notion of an algebraic closure of a field [cf. the discussion of Example 3.8.1], which is not constrained to be a specific set constructed from the field; (DrInv) various categorical constructions such as direct and inverse limits [i.e., such as fiber products of schemes] that are defined by means of some sort of universal property, and which are not constrained to be specific sets even when the given direct or inverse systems are specified set-theoretically; 132 SHINICHI MOCHIZUKI (HomRs) various constructions of (co)homology modules in homological al- gebra that depend on the use of resolutions that satisfy certain abstract properties, but which are not constrained in a strict set-theoretic sense even if the original objects resolved by such resolutions are specified set- theoretically. That is to say, in each of the classical constructions, the “output object” is, strictly speaking, from the point of view of mathematical foundations, not well-defined as a particular set, but rather as a collection of sets [where we note that, typically, this “collection” is not a set!] that are related to one another and hence, in common practice, identified with one another, in the fashion of (ExtInd2)! via unique [modulo, say, some sort of well-defined indeterminacy] isomorphisms by means of some sort of “universal” property. In this context, it is also important to note that, from a foundational point of view, the sort of “(sub)quotient” obtained by applying (ExtInd2) [cf. the discussion of “(sub)quotients” in (sQLTL) and indeed throughout §3.6] must be regarded, a priori, as a formal (sub)quotient, i.e., as some sort of diagram of arrows. That is to say, at least from an a priori point of view, (NSsQ) any explicit construction of a “naive set-theoretic (sub)quotient” necessarily requires the use of some sort of set-theoretic enumeration of each of the individual [set-theoretic] objects that are identified, up to isomorphism, via an application of (ExtInd2). On the other hand, as is well-known, typically such set-theoretic enumerations which often reduce, roughly speaking, to consideration of the “set of all sets”! lead immediately to a contradiction. Indeed, it is precisely this aspect of the constructions of inter-universal Teichmüller theory that motivated the author to include the discussion of species in [IUTchIV], §3. Finally, we recall cf. also the discussion of §3.10 [especially, (Stp7)] below that (LVsQ) it is only in the final portion of inter-universal Teichmüller theory, i.e., once one obtains a formal (sub)quotient that forms a “closed loop”, that one may pass from this formal (sub)quotient to a “coarse/set-theoretic (sub)quotient” by taking the log-volume [cf. the discussion of [Alien], §3.11, (v); [IUTchIII], Remark 3.9.5, (ix); Steps (x), (xi) of the proof of [IUTchIII], Corollary 3.12]. §3.9. Passage and descent to underlying structures One fundamental aspect of inter-universal Teichmüller theory lies in the use of numerous functorial algorithms that consist of the construction input data  output data of certain output data associated to given input data. When one applies such func- torial algorithms, there are two ways in which the output data may be treated LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 133 [cf. [Alien], §2.7, (iii); the discussion of “post-anabelian structures” in [IUTchII], Remark 1.11.3, (iii), (v); [IUTchIII], Remark 1.2.2, (vii)] : (UdOut) One may consider the output data independently of the given input data and functorial algorithms used to construct the output data. In this case, the output data may be regarded as a sort of “underlying structure” associated to the input data. (InOut) One may consider the output data as data equipped with the additional structure constituted by the input data, together with the functorial algo- rithm that gave rise to the output data by applying the algorithm to the input data. Typical examples of this phenomenon in inter-universal Teichmüller theory are the following [cf. the notational conventions of [IUTchI], Definition 3.1, (e), (f)]: (sQGOut) Functorial algorithms that associate to Π v [where v V non ] some sub- quotient group of Π v , such as, for instance, the quotient Π v  G v : In this sort of situation, treatment of the output data [i.e., subquotient group of Π v ] according to (InOut) is indicated by a “(Π v )” following the notation for the particular subquotient under consideration; by contrast, treatment of the output data [i.e., subquotient group of Π v ] according to (UdOut) is indicated by the omission of this “(Π v )”. (MnOut) Functorial algorithms that associate to Π v [where v V non ] some sort of [abelian] monoid equipped with a continuous action by Π v , such as, for instance, [data isomorphic to] various subquotient monoids [i.e., “O  ”, × “O × ”, “O ×μ ”, etc.] of the multiplicative monoid F v : In this sort of situation, treatment of the output data [i.e., monoid equipped with an action by Π v ] according to (InOut) is indicated by a “(Π v )” following the notation for the particular monoid equipped with an action by Π v under consideration; by contrast, treatment of the output data [i.e., monoid equipped with an action by Π v ] according to (UdOut) is indicated by the omission of this “(Π v )”. (PSOut) Functorial algorithms that associate some sort of prime-strip to some sort of input data: In this sort of situation, treatment of the output data [i.e., some sort of prime-strip] according to (InOut) is indicated by a “(−)” [where “−” is the given input data] following the notation for the particular prime-strip under consideration; by contrast, treatment of the output data [i.e., some sort of prime-strip] according to (UdOut) is indicated by the omission of this “(−)”. Perhaps the most central example of (PSOut) in inter-universal Teichmüller theory is the notion of the “q-/Θ-intertwinings” on an F ×μ -prime-strip [cf. the discussion of [Alien], §3.11, (v); [IUTchIII], Remark 3.9.5, (viii), (ix); [IUTchIII], Remark 3.12.2, (ii)]: (ItwOut) This terminology refers to the treatment of the F ×μ -prime-strip ac- cording to (InOut), relative to the functorial algorithm for constructing the q-pilot F ×μ -prime-strip [in the case of the “q-intertwining”] or the Θ-pilot F ×μ -prime-strip [in the case of the “Θ-intertwining”] from some Θ ±ell NF- or D-Θ ±ell NF-Hodge theater. 134 SHINICHI MOCHIZUKI In any situation in which one considers a construction from the point of view of (UdOut) that is to say, as a construction that produces “underlying data” [i.e., “output data”] from “original data” [i.e., “input data”]  input data output data || || original data underlying data it is natural to consider the issue of descent to [a functorial algorithm in] the underlying data of a functorial algorithm in the original data. Here, we say that a functorial algorithm Φ in the original data descends to a functorial algorithm Ψ in the underlying data if there exists a functorial isomorphism Φ Ψ| original data between Φ and the restriction of Ψ, i.e., relative to the given construction original data  underlying data. That is to say, roughly speaking, to say that the functorial algorithm Φ in the original data descends to the underlying data means, in essence, that although the construction constituted by Φ depends, a priori, on the “finer” original data, in fact, up to natural isomorphism, it only depends on the “coarser” underlying data. One elementary example of the phenomenon of descent may be seen in the situation discussed in (HomRs) in §3.8: (HmDsc) The various constructions of (co)homology modules in homological algebra are, strictly speaking, constructions that require as input data not just some given module [whose (co)homology is computed by the construc- tion], but also some sort of resolution of the given module that satisfies certain properties. In fact, however, such constructions of (co)homology modules typically descend, up to unique isomorphism, to constructions whose input data consists solely of the given module. Another illustrative elementary example of the phenomenon of descent is the fol- lowing: Example 3.9.1: Categories of open subschemes. topological space. Write Let X be a scheme, T a · |X| for the underlying topological space of X, · Open(X) for the category of open subschemes of X and open immersions over X, · Open(T ) for the category of open subsets of T and open immersions over T . Then the functorial algorithm X Open(X) LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 135 defined, say, on the category of schemes and morphisms of schemes is eas- ily verified to descend, relative to the construction X  |X|, to the functorial algorithm T Open(T ) defined, say, on the category of topological spaces and continuous maps of topo- logical spaces. That is to say, one verifies immediately that there is a natural functorial isomorphism Open(X) Open(|X|) [i.e., in this case, following the conventions employed in inter-universal Teichmüller theory, a natural functorial isomorphism class of equivalences of categories cf. the discussion of “Monoids and Categories” in [IUTchI], §0]. On the other hand, perhaps the most fundamental example, in the context of inter-universal Teichmüller theory, of this phenomenon of descent is the following [cf. the notational conventions of [IUTchI], Definition 3.1, (e), (f)]: (MnDsc) The topological multiplicative monoid determined by the topological ring given by [the union with {0} of] O  X ) [cf. [Alien], Example 2.12.3, (iii)] that is to say, a construction that, a priori, from the point of view of [AbsTopIII], Theorem 1.9; [AbsTopIII], Corollary 1.10, is a functorial algorithm in the topological group Π X [i.e., “Π v ”, from the point of view (sQGOut)] in fact descends [cf. the discussion at the beginning of [Alien], §2.12; the discussion of [Alien], Example 2.12.3, (i)], relative to passage to the underlying quotient group discussed in (SQGOut), to a functorial algorithm in the topological group G k [i.e., “G v ”, from the point of view (sQGOut)]. Finally, we remark that often, in inter-universal Teichmüller theory, the out- put data of the functorial algorithm Φ of the above discussion is regarded “stack- theoretically”. That is to say, the output data is not a single “set-theoretic object”, but rather a collection [which is not necessarily a set!] of set-theoretic objects linked by uniquely determined poly-isomorphisms of some sort. Typically, this sort of situation arises when one applies (ExtInd2) cf. the discussion of (NSsQ) in §3.8. The most central example of this phenomenon in inter-universal Teichmüller theory is the multiradial algorithm and, especially, the portion of the multi- radial algorithm that involves the log-Kummer-correspondence and closely related operations of Galois evaluation which plays the role of exhibiting the Frobenius-like Θ-pilot as one possibility within a collec- tion of possibilities constructed via anabelian algorithms from étale-like data 136 SHINICHI MOCHIZUKI [cf. the discussion at the end of §3.6, as well as the discussion of §3.10, §3.11, be- low]. That is to say, the log-Kummer-correspondence and closely related operations of Galois evaluation exhibit the Frobenius-like Θ-pilot as one possibility within a collection of possibilities constructed via anabelian algorithms from étale-like data not in a set-theoretic sense [i.e., one possibility/element contained in a set of possibilities], but rather in a “stack-theoretic sense”, in accordance with various applications of (ExtInd2) [cf. the discussion at the end of §3.6], i.e., as one possibility, up to isomorphism, within some [not necessarily set- theoretic!] collection of possibilities. As discussed in (LVsQ) in §3.8, one arrives at a set-theoretic situation i.e., one possibility/element contained in a set of possibilities only after one obtains a “closed loop”, which allows one to pass to a “coarse/set-theoretic (sub)quotient” by taking the log-volume. §3.10. Detailed description of the chain of logical AND relations We begin the present §3.10 with the following well-known and, in some sense, essentially tautological observation: Just as every form of data i.e., ranging from text files and webpages to audiovisual data that can be processed by a computer can, ultimately, be expressed as a [perhaps very long!] chain of “0’s” and “1’s”, the well-known functional completeness, in the sense of propositional calculus, of the collection of Boolean operators consisting of logical AND “∧”, logical OR “∨”, and negation “¬” motivates the point of view that one can, in principle, express the essential logical structure of any mathematical argument or theory in terms of elementary logical relations, i.e., such as logical AND “∧”, logical OR “∨”, and negation “¬”. Indeed, it is precisely this point of view that formed the central motivation and conceptual starting point of the exposition given in the present paper. From the point of view of the correspondence with the terminology and modes of expression that actually appear in [IUTchI-III] and [Alien], the representation given in the present paper of the essential logical structure of inter-universal Te- ichmüller theory in terms of elementary logical relations, i.e., such as logical AND “∧” and logical OR “∨”, may be understood as follows: · Logical AND “∧” corresponds to such terms as · simultaneous execution and · gluing [cf. [IUTchIII], Remark 3.11.1, (ii); [IUTchIII], Remark 3.12.2, (ii), (c itw ), (f itw ); the final portion of [Alien], §3.7, (i); [Alien], §3.11, (iv)]. · Logical OR “∨” corresponds to such terms as · indeterminacies, · poly-morphisms, and · projection/(sub)quotient/splitting [cf. §3.7; the title of [IUTchIII]; [IUTchIII], Remark 3.9.5, (xiii), (ix); [Alien], §3.11, (v); [Alien], §4.1, (iv)]. LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 137 Recall the essential logical structure of inter-universal Teichmüller theory sum- marized in (∧(∨)-Chn) A B = A (B 1 B 2 . . . ) =⇒ A (B 1 B 2 . . . B 1  B 2  . . . ) =⇒ A (B 1 B 2 . . . B 1  B 2  . . . B 1  B 2  . . . ) .. . [cf. the discussion of §3.6, §3.7]. Observe that if the description of the various “possibilities” related via “∨’s” in the above displays is suitably formulated, i.e., without superfluous overlaps, then in fact these logical OR “∨’s” may be understood ˙ as logical XOR ∨’s”, i.e., we conclude the following: ˙ (∧( ∨)-Chn) The essential logical structure of inter-universal Teichmüller theory may be summarized as follows: A B =⇒ A (B 1 ˙ B 2 ˙ . . . ) A (B 1 ˙ B 2 ˙ . . . ˙ B 1  ˙ B 2  ˙ . . . ) =⇒ A (B 1 ˙ B 2 ˙ . . . ˙ B 1  ˙ B 2  ˙ . . . ˙ B 1  ˙ B 2  ˙ . . . ) = .. . Here, we observe the following: ˙ (∧( ∨)-Chn1) The “∧’s” in the above display · arise from the Θ-link, which may be thought of as a re- lationship between certain portions of the multiplica- tive structures of the ring structures arising from the ±ell NF-)Hodge theaters in the domain and codomain of the Θ-link that are common [cf. “∧”!] to these ring structures. This situation is reminiscent of · the fact that from the point of view of Boolean alge- bras, “∧” corresponds to the multiplicative structure of the field F 2 , which may be regarded, via the splitting determined by Teichmüller representatives, as a multi- plicative structure that is common [cf. “∧”!] to Z and F 2 [cf. Example 2.4.6, (iii)], as well as of · the discussion of [Alien], §3.11, (iv), (2 and ), concern- ing the interpretation of the discussion of crystals in [Alien], §3.1, (v), (3 KS ), in terms of the logical rela- tor “∧”, i.e., as objects that may be simultaneously interpreted, up to isomorphism, as pull-backs via one projection morphism and [cf. “∧”!] as pull-backs via the other projection morphism. 138 SHINICHI MOCHIZUKI ˙ ˙ (∧( ∨)-Chn2) The ∨’s” in the above display may be understood as corre- sponding to · various indeterminacies that arise mainly from the log- Kummer-correspondence, i.e., from sequences of it- erates of the log-link, which may be thought of as a device for constructing additive log-shells. The addi- tive structures of the ring structures arising from the ±ell NF-)Hodge theaters in the domain and codomain of the Θ-link are structures which, unlike the corre- sponding multiplicative structures, are not common ˙ [cf. ∨”!] to these ring structures in the domain and codomain of the Θ-link. This situation is reminiscent of · the fact that from the point of view of Boolean al- ˙ corresponds to the additive structure of gebras, ∨” the field F 2 , which is an additive structure that is not ˙ shared [cf. ∨”!], relative to the splitting determined by Teichmüller representatives, by Z and F 2 [cf. Exam- ple 2.4.6, (iii)], as well as of · the discussion of [Alien], §3.11, (iv), (2 and ), concerning the interpretation of the discussion of crystals in [Alien], §3.1, (v), (3 KS ) in terms of the logical relator “∧”, i.e., where we recall that the two pull-backs of the rank one Hodge subbundle [cf. [Alien], §3.1, (v), (5 KS ); the discussion of Hodge structures in [IUTchI], §I2] do ˙ not, in general, coincide [cf. ∨”!], but rather differ by an additive “deformation discrepancy”, namely, the Kodaira-Spencer morphism. ˙ ˙ ˙ (∧( ∨)-Chn3) Taken together, (∧( ∨)-Chn1) and (∧( ∨)-Chn2) may be under- ˙ stood as expressing the fact that the ∨’s” and “∧’s” of the above display correspond, respectively, to the two underlying com- binatorial dimensions i.e., addition and multiplication of a ring or, alternatively, to the two-dimensional nature of the log-theta-lattice [cf. the discussion of [IUTchIII], Remark 3.12.2, (i); the latter portion of [Alien], §3.3, (ii)]. Thus, these two dimensions may be understood, alternatively, as correspond- ing to · the arithmetically intertwined Θ-link and log- link of inter-universal Teichmüller theory, which give rise to the multiradial representation up to suitable ˙ indeterminacies [cf. “∧( ∨)”!] of the Θ-pilot; · the description given in Example 2.4.6, (iii), of the carry-addition operation on the truncated ring of Witt ˙ [cf. “∧( ∨)”!]; ˙ vectors Z/4Z in terms of “∧” and ∨” LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 139 · the filtered crystal discussed in [Alien], §3.1, (v), (5 KS ), where one may think of the filtration [i.e., rank one Hodge subbundle] as “being well-defined up to inde- ˙ terminacies” [cf. “∧( ∨)”!], i.e., up to a “discrepancy”, which is given by the Kodaira-Spencer morphism. ˙ ˙ (∧( ∨)-Chn4) The two dimensions discussed in (∧( ∨)-Chn3) may be under- stood as corresponding to the two dimensions i.e., · the successive iterates of the Frobenius mor- phism in positive characteristic and · successive extensions to mixed characteristic of a ring of Witt vectors [cf. the discussion of the latter por- tion of [Alien], §3.3, (ii)]. This relationship to the two dimensions of a ring of Witt vectors is entirely consistent with the way in which truncated rings of Witt vectors occur in the discussion of Example 2.4.6, (iii), i.e., with the expression ¨ = (∧, ∨) ˙ of mixed characteristic “carry-addition” as a sort of “inter- twining” between addition and multiplication in the field F 2 obtained by “stacking” multiplication “∧” on top of addition ˙ ∨”. Moreover, in this context, we note that the various correspondences observed in ˙ ˙ (∧( ∨)-Chn3) and (∧( ∨)-Chn4) are particularly fascinating in that they assert that the “arithmetic intertwining” in a ring between addition and multiplication i.e., the mathematical structure which is in some sense the main object of study in inter-universal Teichmüller theory may be elucidated by means of a the- ory [i.e., inter-universal Teichmüller theory!] whose essential logical structure, ˙ when written symbolically in terms of Boolean operators such as “∧” and ∨”, ¨ = ∨) ˙ in Example amounts precisely to the description [cf. the discussion of ( 2.4.6, (iii)] of the “Boolean intertwining” that appears in Boolean carry-addition ¨ between Boolean addition ∨” ˙ and Boolean multiplication “∧”. Put another way, it is as if (TrHrc) the arithmetic intertwining which is the main object of study in inter- universal Teichmüller theory somehow “induces”/“is reflected in” a sort of “structural carry operation”, or “trans-hierarchical similitude”, to the Boolean intertwining that constitutes the essential logical struc- ture of the theory [i.e., inter-universal Teichmüller theory] that is used to describe it: arithmetic intertwining  Boolean intertwining! Finally, we observe that it is also interesting to note that the essential mechanism underlying the Kummer theory of theta functions which plays a central 140 SHINICHI MOCHIZUKI role in inter-universal Teichmüller theory, i.e., in inducing the trans-hierarchical similude discussed in (TrHrc) namely, the correspondence Kummer theory of theta functions ←→ one valuation/cusp Kummer theory of algebraic rational functions ←→ multiple valuations/cusps discussed in [IUTchIII], Remark 2.3.3, (viii), (ix), may itself be thought of as a sort of trans-hierarchical similitude between number fields/local fields and function fields over number fields/local fields. At a more technical level, the essential logical structure of inter-universal Te- ˙ ichmüller theory summarized symbolically in (∧( ∨)-Chn) may be understood as consisting of the following steps: (Stp1) log-Kummer-correspondence and Galois evaluation: This step con- sists of exhibiting the Frobenius-like Θ-pilot at the lattice point (0, 0) of the log-theta-lattice i.e., the data that gives rise to the F ×μ -prime-strip that appears in the domain of the Θ-link as one possibility within a collection of possibilities [cf. (ExtInd1)!] constructed via anabelian algorithms from holomor- phic [relative to the 0-column] étale-like data. In this context, it is perhaps worth mentioning that it is a logical tautology that the content of the above display may, equivalently, be phrased as follows: this step consists of the negation “¬” of the assertion of the non-existence of the Frobenius-like Θ-pilot at the lattice point (0, 0) of the log- theta-lattice i.e., the data that gives rise to the F ×μ -prime- strip that appears in the domain of the Θ-link within the col- lection of possibilities constructed via certain anabelian algo- rithms from holomorphic [relative to the 0-column] étale-like data [cf. also the discussion of (RcnLb) below]. At the level of labels of lat- tice points of the log-theta-lattice, this step corresponds to the descent operation (0, 0)  (0, ◦) [cf. the discussion at the end of §3.6; the discussion at the end of §3.9; [IUTchIII], Remark 3.9.5, (viii), (sQ1), (sQ2); [IUTchIII], Theorem 3.11, (ii), (iii)]. Finally, we recall that this step already involves the introduc- tion of the (Ind3) indeterminacy, which may be understood as a sort of coarse algorithmic approximation of the complicated apparatus constituted by the log-Kummer-correspondence and Galois evaluation [cf. (ExtInd1), as well as the discussion of (logORInd) in §3.11 below; the discussion of the algorithmic parallel transport (APT) property in [IUTchIII], Remark 3.11.1, (iv)]. (Stp2) Introduction of (Ind1): This step consists of observing that LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY the anabelian construction algorithms of (Stp1) in fact descend to i.e., are equivalent to algorithms that only require as in- put data the weaker data constituted by [cf. the discussion of “descent” in §3.9] the associated mono-analytic étale-like data, i.e., in the notation of (sQGOut), the “G v ’s”. At the level of labels of lattice points of the log-theta-lattice, this step corresponds to the descent operation (0, ◦)  (0, ◦)  [cf. [IUTchIII], Remark 3.9.5, (viii), (sQ1), (sQ2); [IUTchIII], Theorem 3.11, (i), as well as the references to [IUTchIII], Theorem 3.11, (i), in [IUTchIII], Theorem 3.11, (iii)]. Finally, we recall that this step involves the introduction of the (Ind1) indeterminacy, which [very mildly! cf. the discussion of (Ind3>1+2) in §3.11 below] increases the collection of possibilities under consideration [cf. (ExtInd1)]. (Stp3) Introduction of (Ind2): This step consists of observing that the anabelian construction algorithms of (Stp2) in fact descend to i.e., are equivalent to algorithms that only require as in- put data the weaker data constituted by [cf. the discussion of “descent” in §3.9] the associated mono-analytic Frobenius- like data, i.e., in the notation of (sQGOut) and (MnOut), the “G v  O F ×μ ’s”. v [That is to say, one constructs log-shells, for instance, as submonoids of “O F ×μ ”, as opposed to subquotients of “G v ”.] At the level of labels of v lattice points of the log-theta-lattice, this step corresponds to the descent operation (0, ◦)   (0, 0)  [cf. [IUTchIII], Remark 3.9.5, (viii), (sQ1), (sQ2); [IUTchIII], Theorem 3.11, (i), as well as the references to [IUTchIII], Theorem 3.11, (i), in [IUTchIII], Theorem 3.11, (iii)]. Since the Θ-link may be thought of as a sort of equivalence of labels (0, 0)  ⇐⇒ (1, 0)  i.e., corresponding to the full poly-isomorphism of F ×μ -prime-strips constituted by the Θ-link this descent operation means that the algo- rithm under consideration may be regarded as an algorithm whose input data is the mono-analytic Frobenius-like data (1, 0)  arising from the codomain of the Θ-link. This step involves the introduction of the (Ind2) indeterminacy, which [very mildly! cf. the discussion of (Ind3>1+2) in §3.11 below] increases, at least from an a priori point of view, the collection of possibilities under consideration [cf. (ExtInd1)]. Finally, we recall that this step plays the important role of 141 142 SHINICHI MOCHIZUKI isolating the log-link indeterminacies in the domain [i.e., the (Ind3) indeterminacy of (Stp1)] and the codomain [i.e., the log- shift adjustment discussed in (Stp7) below] of the Θ-link from one another [cf. the discussion of [IUTchIII], Remark 3.9.5, (vii), (Ob7-2); [Alien], §3.6, (iv)]. Here, we recall [cf. the discussion of the final portion of [Alien], §3.3, (ii)] that these log-link indeterminacies on either side of the Θ-link may be understood, in the context of the discussion of (InfH) in §3.3, as corresponding to the copies “C × on either side of the double coset space × “C × \GL + 2 (R)/C ”. (Stp4) Passage to the holomorphic hull: The passage from the collection of possible regions that appear in the output data of (Stp3) to the collection of regions contained in the holomorphic hull relative to the 1-column of the log-theta-lattice of the union of possible regions of the output data of (Stp3) [cf. [IUTchIII], Remark 3.9.5, (vi); [IUTchIII], Remark 3.9.5, (vii), (Ob5); [IUTchIII], Remark 3.9.5, (viii), (sQ3)] is a simple, straighforward application of (ExtInd1), that is to say, of increasing the ˙ set of possibilities [i.e., of ∨’s”]. The purpose of this step, together with (Stp5) below, is to pass from arbitrary regions to regions corresponding to arithmetic vector bundles [cf. [IUTchIII], Remark 3.9.5, (vii), (Ob1), (Ob2)]. (Stp5) Passage to hull-approximants: This step consists of passing from the collection of arbitrary regions contained in the holomorphic hull of (Stp4) to hull-approximants, i.e., regions that have the same global log-volume as the original “arbitrary regions”, but which correspond to arithmetic vector bundles [cf. [IUTchIII], Remark 3.9.5, (vii), (Ob6); [IUTchIII], Remark 3.9.5, (viii), (sQ3)]. This operation does not affect the logical “∧/∨” structure of the algorithm since this operation of passing to hull- approximants does not affect the collection of possible value group portions i.e., “F  -prime-strips” of F ×μ -prime-strips determined by forming the log-volume of these regions [cf. the discussion of [IUTchIII], Remark 2.4.2; the discussion of (IPL) in [IUTchIII], Remark 3.11.1, (iii)]. (Stp6) Passage to a suitable positive rational tensor power of the de- terminant: This step consists of passing from the [regions corresponding to] arithmetic vector bundles obtained in (Stp4), (Stp5) to a suitable tensor power root of a tensor power of the determinant arithmetic line bundle of such an arithmetic vector bundle [cf. [IUTchIII], Remark 3.9.5, (vii), (Ob3), (Ob4); [IUTchIII], Remark 3.9.5, (viii), (sQ3)]. Just as in the case of (Stp5), this operation does not affect the logical “∧/∨” structure of the algorithm since this operation of passing to a suitable positive rational tensor power of the determinant does not affect the collection of possible value group portions LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY i.e., “F  -prime-strips” of F ×μ -prime-strips determined by forming the log-volume of these regions [cf. the discussion of [IUTchIII], Remark 2.4.2; the discussion of (IPL) in [IUTchIII], Remark 3.11.1, (iii)]. 1- column .. . 1- column .. . log log log log log log .. . .. . (Stp7) Log-shift adjustment: The arithmetic line bundles that appear in (Stp6) occur with respect to the arithmetic holomorphic structure i.e., in effect, ring structure at the label (1, 1) of the log-theta-lattice, i.e., at a label vertically shifted by +1 relative to the label (1, 0) that forms the codomain of the Θ-link [cf. the discussion of [IUTchIII], Remark 3.9.5, (vii), (Ob8); [IUTchIII], Remark 3.9.5, (viii), (sQ4)]. That is to say, by applying the algorithm discussed in (Stp1) (Stp6) at each lattice point (1, m) [where m Z] of the 1-column of the log-theta-lattice, we obtain algorithms with input data at (1, m) and output data at (1, m + 1) cf. the diagonal arrows of the diagram shown above. Thus, the totality of all of these diagonal arrows may be thought of as a sort of endomorphism of the 1-column of the log-theta- lattice, i.e., an algorithm whose input data is the 1-column of the log-theta-lattice, and whose output data lies in the same 1-column of the log-theta-lattice. Indeed, one may think of the input data of the algorithm discussed in (Stp1) (Stp6) applied at the lattice point (1, m) as being equipped with the label (1, m)  , while the output data of this algorithm applied at the lattice point (1, m) as being equipped with the label  (1, m + 1), i.e., where one regards these labels (1, m)  and  (1, m + 1) as being “refinements” of the respective original labels (1, m) and (1, m + 1) of the 1-column, which may be recovered by forgetting the additional data “” constituted by the input/output of the algorithm under consideration. Moreover, one may 143 144 SHINICHI MOCHIZUKI consider a “composite refinement”  (1, m)  , i.e., obtained by taking the [q- pilot F ×μ -prime-strip with the same label as the] output data of  (1, m) as the input data of (1, m)  . Here, we note that these input/output labels “” are, in effect, implicit in the species-theoretic sense [cf. the discussion surrounding (NSsQ) in §3.8; the discussion of [IUTchIV], §3] where we observe that the “package of data” constituted by a species may be understood as a sort of label! in which the terms “input data”/“output data” are used throughout [IUTchIII] in the discussion of the multiradial algorithm of [IUTchIII], Theorem 3.11. Then considering the totality of all the diagonal arrows corresponds to considering the “diagonal” [i.e., in the sense of “symmetrized”] collection of data, relative to the symmetry given by Z  m m + 1 Z, which induces compatible symmetries  (1, m)   (1, m + 1)  ; (1, m) (1, m + 1) [cf. the discussion of (Stp8) below]. Here, it is important to observe that the use of these labels (1, −)  ,  (1, −),  (1, −)  renders explicit the sense in which the gluing, up to suitable indeterminacies, arising from the [algorithm denoted by the] corresponding diagonal arrow of the diagram shown above between the input data (1, m)  and the output data  (1, m + 1) is not (!) a gluing embedded in some familiar ambient space [cf. the discusssion of (FxEuc), (FxFld) in the final portion of §3.1], but rather a formal, diagram-theoretic gluing between data with distinct labels [i.e., induced, in effect, by the labels constituted by the various coordinates of the log-theta-lattice, via the various descent opera- tions that appear in the algorithm discussed in (Stp1) (Stp6)], hence, in particular, does not give rise to any nontrivial set-theoretic conclusions i.e., such as the manifest contradiction (!) that arises, if one arbitrarily eliminates/forgets the input/output labels “”, between the gluing constituted by the diagonal arrows of the diagram shown above [i.e., between the local value group portions of the q-pilot codomain F ×μ -prime-strips and domain Θ-pilot F ×μ -prime-strips of the arrows, where the latter is subject to suitable indeterminacies] and the gluings of adjacent local value groups/unit groups constituted by the log-links at any of the intermediate steps that appear in the course of the execution of (Stp1) (Stp6). That is to say, it is only by applying the symmetrization procedure de- scribed above that we obtain a closed loop [cf. the discussion of Example 3.1.1, (iii); the discussion below of (DstMp), (FxGl), (NoCmpIss), (En- glf); the discussion of [IUTchIII], Remark 3.9.5, (ix); [Alien], §3.11, (v)], i.e., in the language of the discussion surrounding (DltLb) in §3.11 below, a situation that simulates via the introduction of suitable indeter- minacies [cf. the discussion of (Stp8) below] a situation in which the LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY distinct labels on the domain and codomain of the Θ-link have been eliminated, hence allows one to draw, in an essentially formal manner [cf. the discussion of (Stp8) below], nontrivial set-theoretic conclu- sions [cf. the discussion surrounding (NSsQ), (LVsQ) in the final portion of §3.8]. Here, we note that the diagonally symmetrized local value groups that one must consider in order to obtain such a closed loop are, in effect, obtained by pulling back these local value groups to the labels  (1, m)  from the original labels “(1, m)” of the 1-column of the log-theta-lattice, i.e., via the forgetting operation  (1, m)   (1, m), hence, in particular, are necessarily subject to the condition of compatibility with the glu- ings of adjacent local value groups/unit groups constituted by the log-links. This compatibility is established by passing to log-volumes and applying the invariance of the log-volume with respect to the log-link [cf. the discussion of [IUTchIII], Remark 3.9.5, (vii), (Ob9)], which may be inter- preted as asserting, in effect, that the log-link induces, via passage to the log-volume, an isomorphism between corresponding adjacent local value groups in the 1-column of the log-theta-lattice [cf. (Stp8) below]. Put another way, this compatibility of the log-volume with the log-links in the 1-column of the log-theta-lattice may be regarded as a sort of “version/analogue for iterates of the log-links in the 1-column” of the saturation with respect to iterates of the log-links in the 0-column discussed in (logORInd) [cf. §3.11] below, hence, in particular, as a sort of “1-column version/analogue” of the re- markable phenomenon constituted by the “stark contrast between the po- tency of [the 0-column] (logORInd) and the utterly meaningless nature of (ΘORInd)” [cf. the discussion of §3.11 below]. Finally, in this context, it is interesting to observe, from a historical point of view, that the set-theoretic confusion [e.g., in the form the manifest con- tradiction discussed above!] that arises at intermediate steps in the course of the execution of (Stp1) (Stp6) if one does not take into account the various labels of the log-theta-lattice, as well as the additional input/output labels discussed above [i.e., which are, in effect, induced by various labels or collections of labels of the log-theta-lattice ], is remarkably reminiscent of the his- torical Weierstrass/Riemann dispute that arose in complex function theory in the context of the theory of analytic con- tinuation/Riemann surfaces, i.e., prior to the development of modern axiomatic set theory in the early 20-th century [cf. the discussion of §1.5] where we note that, in this analogy, the various labels of the log-theta-lattice, as well as the additional input/output labels discussed above, correspond to the set-theoretically distinct labels of various copies of the complex open unit disc that appear in the theory of analytic continuation/Riemann surfaces. (Stp8) Passage to log-volumes: The closed loop of (Stp7) [cf. also the discussion of Example 3.1.1, (iii); the discussion below of (DstMp), (FxGl), (NoCmpIss), (Englf)] implies that the crucial logical AND “∧” relation 145 146 SHINICHI MOCHIZUKI carefully maintained throughout the execution of (Stp1) (Stp7) yields, upon taking the log-volume, a logical AND “∧” relationship between the original q-pilot input F  -prime-strip and a certain algorithmically constructed collection of possible output F  -prime-strips within the same container, i.e., some copy of the real numbers “R” [cf. [IUTchIII], Remark 3.9.5, (vii), (Ob9); [IUTchIII], Remark 3.9.5, (viii), (sQ5); [IUTchIII], Remark 3.9.5, (ix); the discussion of Substeps (xi-d), (xi-e) of the proof of [IUTchIII], Corollary 3.12; the discussion of [IUTchIII], Remark 3.12.2, (ii); [Alien], §3.11, (v)]. The inequality in the statement of [IUTchIII], Corollary 3.12, then follows as a formal consequence of the invariance of the log-volume with respect to the log- link [cf. the discussion of the final portion of (Stp7) above; the discussion of Substeps (xi-e), (xi-f), (xi-g) of the proof of [IUTchIII], Corollary 3.12; [IUTchIII], Remark 3.12.2, (iv), (v)]. Here, we observe that the various indeterminacies [such as (Ind1), (Ind2), (Ind3)] that arise in the course of applying (Stp1) (Stp7) may be thought of as a sort of monodromy associated to the closed loop of (Stp7) [cf. also the discussion below of (DstMp), (FxGl), (NoCmpIss), (Englf)]. In this context, we recall from (Stp7) that the diagram of arrows from the 1-column to the 1-column in (Stp7) admits symmetries (1, m) (1, m + 1) [where m Z] that are compatible with all of the arrows in the diagram, as well as with the various arrows of the log-Kummer-correspondence in the 1-column. These symmetries allow one to synchronize the various “monodromy indeterminacies” associated to each [i.e., to each application of (Stp1) (Stp6)], so that one may think in terms of a single, synchronized collection of “monodromy indeterminacies” associated to the totality of ’s” in (Stp7). Before proceeding, we pause to examine the meaning of the term “closed loop” in (Stp7), (Stp8), which is sometimes a source of confusion. The intended meaning of this term is that the sequence of mathematics objects on which the series of operations in (Stp1) (Stp6) [cf. also [IUTchIII], Fig. I.8] are performed forms a closed loop in the sense that the ultimate output data lies in the same container [i.e., up to a log-shift in the 1-column, as discussed in (Stp7)] as the input data. On the other hand, at the level of the actual mathematical objects that one is working with, the term “closed loop” has the potential to result in certain fundamental misunderstandings, since it may be [mistakenly!] interpreted as suggesting that (DstMp) one is considering two distinct mappings of abstract prime-strips to [Θ, q-]pilot prime-strips. Once one takes this point of view (DstMp), there is inevitably an issue of diagram commutativity, i.e., the issue discussed in §3.6, (Syp2), that one must contend LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 147 with. As discussed in Example 2.4.5, (iv), (v), (vi), (vii), (viii) [cf. also Examples 3.10.1, 3.10.2 below], this point of view (DstMp) corresponds to EssOR-IUT [i.e., “essentially OR IUT”], which, as the name suggests, is essentially [thought not precisely!] equivalent to OR-IUT, and in particular, constitutes a fundamental misunderstanding of the logical structure of inter-universal Teichmüller theory. Indeed, the chain of AND relations “∧” discussed in §3.6, as well as the present §3.10, which lies at the heart of the essential logical structure of inter- universal Teichmüller theory, consists precisely of (FxGl) fixing the Frobenius-like q-pilot at the lattice point (1, 0), as well as the gluing [i.e., “∧”!] via the Θ-link of this q-pilot at (1, 0) to the Frobenius- like Θ-pilot at the lattice point (0, 0) [cf. [IUTchIII], Remark 3.12.2, (ii), (c itw ), (e itw ), (f itw )]. One then proceeds to add to this Θ-pilot at the lattice point (0, 0) more and more ˙ possibilities/indeterminacies [i.e., “∨”, or, alternatively, ∨”!] in order to obtain data that descends to the same label [i.e., up to a log-shift in the 1-column, as discussed in (Stp7)] as the q-pilot at (1, 0). That is to say, (NoCmpIss) there is never any issue of compatibility between two distinct map- pings of abstract prime-strips, as in (DstMp). From a pictorial point of view, at the level of mathematical objects, one is working in (Stp1) (Stp8) not with a “closed loop” (!), but rather a single fixed line segment = = = = = corresponding to the gluing [i.e., “∧”!] of (FxGl) [so the “•’s” on the left and right correspond, respectively, to the Θ-pilot at (0, 0) and the q-pilot at (1, 0)], which ˙ is then subjected to subsequent “fuzzifications” [i.e., “∨”, or alternatively, ∨”!] of the Θ-pilot at (0, 0) [denoted in the following display by the notation “(. . . )”, which may be thought of as representing a “fuzzy disc” that contains the “•” on the left]  = = = ( =) = = ( = =) = ( = = =) ( = = =     ( = = =  ( = = = = = = = = = = = =) = = =) = = ) that terminate in a situation [cf. the final line of the above display] in which (Englf) the fuzzifications engulf the q-pilot at (1, 0), i.e., a situation in which the distinct labels may be eliminated and nontrivial consequences may be 148 SHINICHI MOCHIZUKI obtained [cf. (Stp7), (Stp8), as well as the discussion surrounding (DltLb) in §3.11 below]. Thus, in summary, throughout the series of fuzzification operations constituted by (Stp1) (Stp8), the line segment representing the gluing [i.e., “∧”!] of (FxGl) remains fixed, so [cf. (NoCmpIss), (Englf)] there is never any issue of compat- ibility between two distinct mappings of abstract prime-strips, as in (DstMp). Example 3.10.1: Symmetries as a fundamental non-formal aspect of glu- ings. One psychological aspect, and indeed [in many cases] possible cause, of the fundamental misunderstandings discussed above [cf. the discussion above sur- rounding (DstMp), as well as the discussion surrounding (RfsDlg), (DngPrc) in §1.12] concerning the essential logical structure of inter-universal Teichmüller the- ory i.e., the [erroneous!] point of view that this essential logical structure of inter-universal Teichmüller theory should be understood as centering around an issue of diagram commutativity is the following. In any sort of gluing sit- uation i.e., from a category-theoretic point of view, any sort of situation [cf. the numerous examples of gluings discussed throughout the present paper, e.g., in Examples 2.3.2, 2.4.1, 2.4.2, 2.4.3, 2.4.7, 2.4.8, 3.3.1, 3.5.2] in which one considers the [possibly formal] inductive limit I of a diagram of the form Y 1 ←− X −→ Y 2 there is a certain tendency to fall into the “mental trap” of believing that it is a “tautology” that (UnvPrp) any conceivable nontrivial/interesting property of I should be understood in terms of the universal property of an inductive limit, i.e., the property to the effect that any arrow from I to some object Z should be understood in terms of the issue of commutativity of a diagram of the form X −→ Y 2  ? Y 1 −→ Z i.e., where the left-hand vertical and upper horizontal arrows are the arrows in the definition of I. Here, we observe that (UnvPrpOR) the diagram commutativity issue that arises when one considers the uni- versal property discussed in (UnvPrp) is essentially a logical OR “∨” situation: that is to say, one can pass from X to Z in the diagram of (UnvPrp) via Y 1 OR via Y 2 , but there is, a priori, no way of passing from such an “OR situation” to the desired “AND situation” that arises when commutativity holds, i.e., the situation where one knows that there is a single arrow from X to Z that is simultaneously equal to the composite of the two arrows that pass through Y 1 and equal to the composite of the two arrows that pass through Y 2 LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 149 cf. the discussion of Example 2.4.5, (iv), (v), (vi), (vii), (viii). Of course, it is indeed a tautology that I satisfies a universal property as in (UnvPrp), but the point is that (FlsUnv) the fact that I satisfies such a universal property does not by any means imply i.e., as one might falsely conclude from the nuance carried by the word “universal” in ordinary, non-technical contexts! that any con- ceivable nontrivial/interesting property of I is best understood in terms of this universal property. That is to say, (SymPrp) there are numerous examples in mathematics of objects “I” that are constructed as gluings, but that satisfy important nontrivial properties such as symmetry properties e.g., of the sort discussed in §3.2 [cf., especially, Example 3.2.2!] that do not admit any natural “general nonsense” formulation in terms of the universal property of (UnvPrp). In the spirit of (UnvPrpOR), it is of interest to note that (SymPrpAND) symmetry properties typically concern some sort of invariant, or “coric”, structure that is commonly shared throughout various “local- izations” of the diagram that gives rise to I, where we recall that it is es- sentially a tautology that this sort of notion of a commonly shared “coric” property is a logical AND “∧” situation cf. the discussion of Example 2.4.5, (iv), (v), (vi), (vii), (viii), as well as the dis- cussion of (FxGl), (NoCmpIss), (Englf), above. Well-known elementary examples of the phenomenon discussed in (SymPrp) include (i) the projective general linear symmetries [i.e., “P GL 2 ”] of the pro- jective line, which may be constructed as a gluing of two copies of the affine line [cf. Example 2.4.7, (iv), (v)]; (ii) the group structure of an elliptic curve, which, as is well-known, may be constructed as a gluing of two cubic planar affine curves; (iii) the group of general linear oriented symmetries GL + 2 (R) of the two- dimensional R-vector space R 2 cf. the discussion of the double coset space × C × \GL + 2 (R)/C in §3.3, (InfH) in the situation of the complex Teichmüller defor- mations discussed in Example 3.3.1, where we recall that this situation may be regarded [cf. the discussion at the beginning of Example 3.5.2] as a “gluing” of two distinct copies of the complex plane C along a common underlying two-dimensional R-vector space R 2 [cf. also Example 3.5.2, (iii)], and we observe that these symmetries GL + 2 (R) allow one to re- gard each of the two holomorphic structures [i.e., copies of C] that appear in this gluing as indeterminate GL + 2 (R)-conjugates of the subgroup + × C GL 2 (R) inside the common container GL + 2 (R); (iv) the common holomorphic structure that appears in the classical theory of analytic continuation of one-variable complex holomorphic 150 SHINICHI MOCHIZUKI functions cf., e.g., the discussion of the theory of analytic continuation of the complex logarithm in the discussion surrounding (FxEuc) in §3.1, as well as in the historical discussion of §1.5 and the discussion of the toral rotations “C × in §3.3, (InfH) where we observe that such analytic continuations may be regarded as gluings of copies of the complex open unit disc, and that the common holomorphic structure may be regarded as a sort of common symmetry of such gluings, i.e., if one thinks in terms of “almost complex structures”, that is to say, in terms of the symmetry of the tangent bundle given by multiplication by i = −1. Here, we observe in passing that it is of interest to note, in the context of (ii), that the group structure of an elliptic curve, as well as the existence of invariant [i.e., with respect to this group structure] differentials on the elliptic curve, play a central role in the argument discussed in Example 3.2.1 an argument that itself played a fundamental role in motivating the essential logical structure of inter-universal Teichmüller theory [cf. Example 3.2.1, (viii)]. Of course, as discussed extensively in §3.3 [cf. (InfH); Examples 3.3.1, 3.3.2], the situations discussed in (iii), (iv) also exhibit numerous important structural similarities to various important aspects of inter-universal Teichmüller theory. Example 3.10.2: Chains of logical AND relations via commutative dia- grams. We maintain the notation of Example 3.10.1. (i) As discussed at the beginning of Example 3.10.1, (ORAch) the “diagram-commutativity”, or “OR approach” [cf. (UnvPrp), (UnvPrpOR)], X −→ Y 2 Y 1 ←−   ?  Z to analyzing the structure of the [possibly formal] inductive limit I of the upper line of the above diagram constitutes a sort of “mental trap” that appears to be, in many cases, one of the main causes of the fundamental misunderstandings that exist in certain sectors of the mathematical community concerning inter-universal Teichmüller theory. [Here, we recall that, in the situation considered in inter-universal Teichmüller theory, the initial gluing in the first horizontal line of the diagram of (ORAch) corresponds to the gluing constituted by the Θ-link between the “Θ-pilot object in the Θ-(Θ ±ell NF-)Hodge theater” Y 1 and the “q-pilot object in the q-(Θ ±ell NF- )Hodge theater” Y 2 along the prime-strip data X, while “Z” is to be understood as some sort of container that contains both Y 1 and Y 2 .] The state of affairs summarized in (ORAch) thus prompts the following question: (Q∧(∨)CCD) Since the tendency of many mathematicians especially those who work in abstract areas of arithmetic geometry! to try to interpret the essential logical structure of inter-universal Teichmüller theory in terms of the [incorrect!] “OR approach” of (ORAch) appears to stem, to a substantial extent, from the fact that such mathematicians often have a LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 151 strong desire to formulate structural properties of mathematical objects in terms of commutative diagrams, is it possible to somehow formulate the essential logical structure of inter-universal Teichmüller theory ˙ summarized in (∧(∨)-Chn) or, alternatively, (∧( ∨)-Chn) in terms of some sort of collection of commutative diagrams? Here, we recall from the discussion at the beginning of the present §3.10 [cf. also the discussion surrounding (SymIUT) in §1.12] that the main thrust of the present paper lies in formulating the essential logical structure of inter-universal Teichmüller theory [modulo certain “blackboxes” in anabelian geometry and the theory of étale theta functions] in terms of logical AND “∧”/logical OR “∨” relations. The fundamental motivation for this approach taken in the present paper lies in the point of view that such logical AND “∧”/logical OR “∨” relations constitute the “most primitive/fundamental/universal” means available for documenting the essential logical structure of a mathematical argument. This point of view is also closely related to the point of view of computer verification of mathematical arguments discussed at the beginning of §1.12 [cf. (CmbVer), (Algor)]. On the other hand, even if this point of view is in some sense “correct” from an abstract, theoretical standpoint, it is not necessarily the case that this point of view is also correct from the somewhat more practical standpoint of developing an optimally efficient means of communicating the essential logical structure of inter-universal Teichmüller theory to other mathematicians. It is precisely this practical standpoint that motivates the question posed in (Q∧(∨)CCD). (ii) In a word, it is not very difficult to give an affirmative answer to the question posed in (Q∧(∨)CCD). Indeed, (∧(∨)CCD) one may formulate the essential logical structure of inter-universal ˙ Teichmüller theory summarized in (∧(∨)-Chn) or, alternatively, (∧( ∨)- Chn) in terms of a chain of [tautologically!] commutative dia- grams as follows: Y 1 ←− X −→ Y 2  ! !  ! ! Y 1  ←− X −→ Y 2  ! !  ! Y 1  .. . ←− X .. . −→ Y 2 .. . ! where the left-hand vertical arrows are natural inclusion morphisms into larger and larger containers, and “ !” denotes the tautological commutativity of the square in question. That is to say, this chain of [tautologically!] commutative diagrams exhibits the initial gluing in the first horizontal line i.e., the gluing constituted by the Θ- link between the “Θ-pilot object in the Θ-(Θ ±ell NF-)Hodge theater” Y 1 and the “q-pilot object in the q-(Θ ±ell NF-)Hodge theater” Y 2 along the prime-strip data X as an object that maps tautologically to the gluings in the subsequent horizontal lines, which consist of copies of the initial gluing in the first horizontal line, but 152 SHINICHI MOCHIZUKI with the “Θ-pilot object in the Θ-(Θ ±ell NF-)Hodge theater” Y 1 regarded up to suitable indeterminacies, or “alternative possibilities” [i.e., as discussed in (Stp1) (Stp8)], exhibited in larger and larger containers “Y 1  ”, “Y 1  ”, . . . . Put another way, this approach consists of embedding Y 1 into larger and larger containers [i.e., the containers obtaining by adding the various indeterminacies] until the container becomes sufficiently large as to contain/engulf [not only Y 1 , but also (!)] Y 2 [cf. the discussion surrounding (Englf)]. (iii) At first glance, the approach discussed in (∧(∨)CCD) may appear to some arithmetic geometers to be entirely unfamiliar and fundamentally different from the approach taken in the numerous theories in arithmetic geometry that existed prior to the appearance of inter-universal Teichmüller theory. In fact, however, the approach discussed in (∧(∨)CCD) is entirely analogous to the approach taken in the classical theory of crystals. Moreover, this analogy with the classical the- ory of crystals is completely compatible with the discussion of the strong structural resemblances between inter-universal Teichmüller theory and the theory of crys- tals given in [Alien], §3.1, (v) [cf. also §3.5 of the present paper, as well as the ˙ discussion of (∧( ∨)-Chn) in the present §3.10]. Indeed, the “container of inde- terminacies” discussed in (ii) may be understood as corresponding to the PD- envelopes/thickenings that apppear in the theory of crystals. That is to say, the approach typically taken in the theory of crystals to constructing [nontrivial!] crystals i.e., to constructing isomorphisms p 1 (−) p 2 (−) between the pull-backs via the two natural projections Y p 1 ←− Y × Y p 2 −→ Y of some object “(−)” on some scheme Y proceeds not by securing some sort of commutative diagram p 1 p 2 Y ←− Y × Y −→ Y   ?  Z i.e., corresponding to the [incorrect!] “OR approach” of (ORAch)! such that the object “(−)” on Y descends to Z, but rather by restricting to the “suf- ficiently large container” constituted by a suitable PD-envelope/thickening of the diagonal in Y × Y and verifying that this container is indeed sufficiently large that the restriction to this PD-envelope/thickening of p 1 (−) already contains, up to isomorphism, the inverse image p −1 2 (−). §3.11. The central importance of the log-Kummer-correspondence In the context of the discussion of §3.10, it is important to recall that, whereas (Stp2) (Stp8) are technically trivial in the sense that they concern operations that are very elementary and only require a few lines to describe, the log-Kummer- correspondence and Galois evaluation operations that comprise (Stp1) depend on the highly nontrivial theory of [EtTh] and [AbsTopIII]. Moreover, the technical LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 153 description of these operations that comprise (Stp1) occupies the bulk of [IUTchI- III]. The central importance of (Stp1) may also be seen in the subordinate nature of (Ind1), (Ind2) [which occur in (Stp2), (Stp3)] relative to (Ind3) [which occurs in (Stp1)], i.e., in the sense that (Ind3>1+2) once one constructs the output of the multiradial representation of the Θ-pilot [cf. [IUTchIII], Theorem 3.11, (ii)] via tensor-packets of log- shells in such a way that each local portion of this output is stable with respect to the indeterminacy (Ind3), these local portions of the output are automatically “essentially stable” [i.e., stable up to discrepancies at the valuations V bad that affect the resulting log-volumes only up to very small/essentially negligible order] with respect to the indeterminacies (Ind1), (Ind2) [cf. [IUTchIII], Theorem 3.11, (i)]. Finally, we observe that this property (Ind3>1+2) is strongly reminiscent of the discussion of (CnfInd1+2) and (CnfInd3) in §3.5. One way to understand the content of the operations of (Stp1) is as follows. These operations may be regarded as a sort of (logORInd) saturation of the Frobenius-like Θ-pilot at the lattice point (0, 0) of the log-theta-lattice i.e., which is linked, via the Θ-link, to the Frobenius-like q-pilot at (0, 1) with respect to all of the possibil- ities that occur in the 0-column of the log-theta-lattice, i.e., all of the possibilities that arise from a possible confusion between the domain and codomain of the log-links in the 0-column [cf. the description of (Stp1)]. In this sense, the content of (Stp1) is formally reminiscent of the “(NeuORInd)” that appeared in the discussion of §3.4, i.e., which may be understood as a sort of (ΘORInd) saturation of the Frobenius-like Θ-pilot at the lattice point (0, 0) of the log-theta-lattice with respect to all of the possibilities i.e., “Θ-plt”, “q-plt” [cf. (NeuORInd2)] that arise from a possible confusion between the domain and codomain of the Θ-link joining the lattice points (0, 0) and (0, 1). [In this context, we note that the logical OR “∨’s” that appear in (logORInd), ˙ (ΘORInd) may in fact be understood as logical XOR ∨’s” cf. the discussion ˙ surrounding (∧( ∨)-Chn) in §3.10.] On the other hand, whereas, as observed in the discussion at the end of §3.4, (ΘORInd) yields a meaningless/useless situation that does not give rise to any interesting mathematical consequences, (logORInd), by contrast, is a highly potent technical device that forms the technical core of inter-universal Teichmüller theory. Before preceding, we observe that, in this context, it is interesting to note that both of these “saturation operations” (logORInd) and (ΘORInd) are in some sense qualitatively similar to the label crushing operation (ExtInd2). Indeed, (ExtInd2) consists, roughly speaking, of regarding mathematical objects of a certain type up to isomorphism, i.e., of saturating within an isomorphism class of mathematical objects of a certain type [cf. the discussion of (ExtInd2) in §3.8, as well as the discussion of (DltLb) below]. 154 SHINICHI MOCHIZUKI The stark contrast between the potency of (logORInd) and the utterly mean- ingless nature of (ΘORInd) is highly reminiscent of the central role played, in Example 3.3.2, (iv), by invariance with respect to  ι =  01 −1 0 C × GL + 2 (R) [where we recall from (InfH) that C × corresponds to the log-link!], which lies in stark contrast to the utterly meaningless nature of considering invariance with respect to dilations λ 0 1 0 GL + 2 (R) [where we recall from (InfH) that such dilations correspond to the Θ-link]. One way to witness the potency of (logORInd) is as follows. Recall that the Θ-link, by definition [cf. [IUTchIII], Definition 3.8, (ii)], consists of · a dilation applied to the local value group portions of the ring structures in its domain and codomain, coupled with · a full poly-isomorphism which preserves log-volumes, hence is non- dilating! between the local “O ×μ ’s”, i.e., the local unit group portions, of these ring structures. By contrast, the log-links in the 0-column of the log-theta-lattice have the effect of “juggling/rotating/permuting” the local value group portions and local unit group portions of the ring structures that appear in this 0-column [cf., e.g., the discussion of [Alien], Example 2.12.3, (v)]. From this point of view, the tautologically vertically coric i.e., invariant with respect to the application of the log-link! nature of the output data of (logORInd) is already somewhat “shocking” in nature. That is to say, the tautologically vertically coric nature of this output data of (logORInd) suggests that (Di/NDi) this output data already exhibits some sort of equivalence, up to per- haps some sort of mild discrepancy, between the dilated and non-dilated portions of the Θ-link. Such an equivalence already strongly suggests that some sort of bound on heights should follow as a formal consequence, i.e., in the style of the classical argument that implies the isogeny invariance of heights of elliptic curves [cf. the discussion of [Alien], §2.3, §2.4, as well as the discussion of Example 3.2.1 in the present paper; the discussion of §3.5 in the present paper]. Finally, we conclude by emphasizing that, in inter-universal Teichmüller theory, (DltLb) ultimately one does want to find some way in which to delete/eliminate the distinct labels on the Θ- and q-pilot objects [i.e., “Θ-plt” and “q-plt”] in the domain and codomain of the Θ-link [cf. the discussion of Example 3.1.1, (iii); the discussion of (AOL4), (AOΘ4) in §3.4; (Stp7), (Stp8) in §3.10], that is to say, not via the naive, simple-minded approach of (ΘORInd) [i.e., (NeuORInd2) in the discussion of §3.4], but rather via the indirect approach of applying descent operations (0, 0) (Stp1)  (0, ◦) (Stp2)  (0, ◦)  (Stp3)  (0, 0)  (Stp3) ⇐⇒ (1, 0)  LOGICAL STRUCTURE OF INTER-UNIVERSAL TEICHMÜLLER THEORY 155 as discussed in (Stp1) (Stp8) [cf., especially, (Stp7), (Stp8)] of §3.10, i.e., an approach that centers around (logORInd). This approach is based on the vari- ous anabelian reconstruction algorithms discussed in (Stp1) (Stp3), which allow one to exhibit the Frobenius-like Θ-pilot object at (0, 0) as one possibil- ity among some broader collection of possibilities that arise from the introduction of various types of indeterminacy. In this context, we observe [cf. the discus- sion of (ExtInd2), (NSsQ) at the end of §3.9] that since such anabelian recon- struction algorithms only reconstruct various types of mathematical objects [i.e., monoids/pseudo-monoids/mono-theta environments, etc.] not “set-theoretically on the nose” [i.e., not in the sense of strict set-theoretic equality], but rather up to [a typically essentially unique, if one allows for suitable indeterminacies] isomorphism, it is not immediately clear (RcnLb) in what sense such anabelian reconstruction algorithms yield a recon- struction of the crucial labels i.e., such as “(0, 0)” that underlie the crucial logical AND “∧” structure discussed in §3.4 [cf., especially, (AOL1), (AOΘ1)]. The point here is that indeed such anabelian reconstruction algorithms are not capable of reconstructing such labels “set-theoretically on the nose”. On the other hand, in this context, it is important to recall the essential substantive content of the various labels involved: (HolFrLb) (0, 0): The holomorphic Frobenius-like data labeled by (0, 0) con- sists of various monoids/pseudo-monoids/mono-theta environments, etc., regarded as abstract monoids/pseudo-monoids/mono-theta environments, etc., i.e., as objects that are not equipped with the auxiliary data of how they might have been reconstructed via anabelian algorithms from holo- morphic étale-like data labeled (0, ◦) [cf. the discussion of (UdOut), (In- Out), (PSOut), (ItwOut) in §3.9]. In particular, such monoids/pseudo- monoids/mono-theta environments, etc., are not invariant with respect to the “juggling/rotating/permuting” of local value group portions and lo- cal unit group portions effected by the log-links in the 0-column of the log-theta-lattice, but rather correspond to a temporary cessation [cf. the label (0, 0) as opposed to the label (0, ◦)!] of this operation of jug- gling/rotation/permutation. (MnAlyLb) (0, 0)  : The mono-analytic Frobenius-like data labeled by (0, 0)  consists of the F ×μ -prime-strip determined by the Frobenius-like Θ- pilot at (0, 0), regarded as an abstract F ×μ -prime-strip [cf. the discus- sion of (UdOut), (InOut), (PSOut), (ItwOut) in §3.9]. Thus, the transition of labels (0, 0)  (0, 0)  consists of an operation of forgetting some sort of auxiliary structure [cf. the discussion of (UdOut) in §3.9]. Here, we recall that this construction of the F ×μ -prime-strip determined by the Frobenius-like Θ-pilot at (0, 0) is technically possible precisely because of the “temporary cessation” discussed above [cf. the discussion of the definition of the Θ-link in [Alien], §3.3, (ii), as well as in §3.3 of the present paper]. 156 SHINICHI MOCHIZUKI Thus, the nontrivial substantive content of the anabelian reconstruction algorithms of (Stp1) (Stp3) and hence of the descent operations (0, 0) (Stp1) (Stp3)  (0, 0)  that result from these anabelian reconstruction algorithms consists of statements to the effect that (FrgInv) the operation of forgetting discussed in (MnAlyLb) can in fact, if one allows for suitable indeterminacies, be inverted. It is precisely this invertibility (FrgInv), up to suitable indeterminacies, of the operation of forgetting discussed in (MnAlyLb), together with the fact that (GluDt) the only data appearing in the reconstruction algorithms [i.e., in the 0-column] that is glued [cf. the discussion of [IUTchIII], Remark 3.11.1, (ii); the final portion of [Alien], §3.7, (i), as well as the discussion of the (Ind2) indeterminacy in the final portion of §3.11, (Stp3), in the present paper] to data in the 1-column is the F ×μ -prime-strip labeled (0, 0)  , that ensures that the descent operations discussed above do indeed preserve the crucial logical AND “∧” relations discussed in §3.4, §3.6, §3.7, §3.10, i.e., even though the reconstruction algorithms underlying these descent operations do not yield reconstructions of the various labels “(0, 0)”, etc., “set-theoretically on the nose”. 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